today: section 8.2, continued. to solve recurrence ... · to solve recurrence relations, the best...
TRANSCRIPT
![Page 1: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/1.jpg)
Today: Section 8.2, continued.
Last time:To solve recurrence relations, the best we can do is make educatedguesses, according to what kind of relation we have.
A linear homogeneous recurrence relation of degree k withconstant coefficients is a recurrence relation of the form
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k,
where c1, c2, . . . , ck are real numbers, and ck ‰ 0.
Solutions to this kind of relation come in the form rn, where r is aroot of the characteristic equation, which is obtained by pluggingin rn, dividing through by rn´k, and solving for 0:
rk ´ c1rk´1 ´ c2r
k´2 ´ ¨ ¨ ¨ ´ ck´1r ´ ck “ 0.
The roots of this equation are called the characteristic roots.
![Page 2: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/2.jpg)
Today: Section 8.2, continued.
Last time:To solve recurrence relations, the best we can do is make educatedguesses, according to what kind of relation we have.
A linear homogeneous recurrence relation of degree k withconstant coefficients is a recurrence relation of the form
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k,
where c1, c2, . . . , ck are real numbers, and ck ‰ 0.
Solutions to this kind of relation come in the form rn, where r is aroot of the characteristic equation, which is obtained by pluggingin rn, dividing through by rn´k, and solving for 0:
rk ´ c1rk´1 ´ c2r
k´2 ´ ¨ ¨ ¨ ´ ck´1r ´ ck “ 0.
The roots of this equation are called the characteristic roots.
![Page 3: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/3.jpg)
Today: Section 8.2, continued.
Last time:To solve recurrence relations, the best we can do is make educatedguesses, according to what kind of relation we have.
A linear homogeneous recurrence relation of degree k withconstant coefficients is a recurrence relation of the form
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k,
where c1, c2, . . . , ck are real numbers, and ck ‰ 0.
Solutions to this kind of relation come in the form rn, where r is aroot of the characteristic equation, which is obtained by pluggingin rn, dividing through by rn´k, and solving for 0:
rk ´ c1rk´1 ´ c2r
k´2 ´ ¨ ¨ ¨ ´ ck´1r ´ ck “ 0.
The roots of this equation are called the characteristic roots.
![Page 4: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/4.jpg)
Example from last time: plugging an “ rn into the recursionrelation
an “ ´an´1 ` an´2 ` an´3
gets
rn “ ´rn´1 ` rn´2 ` rn´3
,
which is true if and only if
0 “ rn ` rn´1 ´ rn´2 ´ rn´3
“ rn´3pr3 ` r2 ´ r ´ 1q
which is true if and only if r “ 0 or
0 “ r3 ` r2 ´ r ´ 1looooooooooomooooooooooon
characteristic equation
“ pr ` 1q2pr ´ 1q.
The characteristic roots are r1 “ 1 (with multiplicity 1) andr2 “ ´1 (with multiplicity 2).
![Page 5: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/5.jpg)
Example from last time: plugging an “ rn into the recursionrelation
an “ ´an´1 ` an´2 ` an´3
gets
rn “ ´rn´1 ` rn´2 ` rn´3,
which is true if and only if
0 “ rn ` rn´1 ´ rn´2 ´ rn´3
“ rn´3pr3 ` r2 ´ r ´ 1q
which is true if and only if r “ 0 or
0 “ r3 ` r2 ´ r ´ 1looooooooooomooooooooooon
characteristic equation
“ pr ` 1q2pr ´ 1q.
The characteristic roots are r1 “ 1 (with multiplicity 1) andr2 “ ´1 (with multiplicity 2).
![Page 6: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/6.jpg)
Example from last time: plugging an “ rn into the recursionrelation
an “ ´an´1 ` an´2 ` an´3
gets
rn “ ´rn´1 ` rn´2 ` rn´3,
which is true if and only if
0 “ rn ` rn´1 ´ rn´2 ´ rn´3
“ rn´3pr3 ` r2 ´ r ´ 1q
which is true if and only if r “ 0 or
0 “ r3 ` r2 ´ r ´ 1looooooooooomooooooooooon
characteristic equation
“ pr ` 1q2pr ´ 1q.
The characteristic roots are r1 “ 1 (with multiplicity 1) andr2 “ ´1 (with multiplicity 2).
![Page 7: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/7.jpg)
Example from last time: plugging an “ rn into the recursionrelation
an “ ´an´1 ` an´2 ` an´3
gets
rn “ ´rn´1 ` rn´2 ` rn´3,
which is true if and only if
0 “ rn ` rn´1 ´ rn´2 ´ rn´3
“ rn´3pr3 ` r2 ´ r ´ 1q
which is true if and only if r “ 0 or
0 “ r3 ` r2 ´ r ´ 1looooooooooomooooooooooon
characteristic equation
“ pr ` 1q2pr ´ 1q.
The characteristic roots are r1 “ 1 (with multiplicity 1) andr2 “ ´1 (with multiplicity 2).
![Page 8: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/8.jpg)
Theorem: Solving linear homogeneous recurrencesLet c1, c2, . . . , ck be real numbers. Suppose that the characteristicequation has roots r1, r2, . . . , r` with multiplicities m1,m2, . . . ,m`.Then a sequence tanunPN is a solution of the recurrence relationan “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k if and only if
an “ p1pnqrn1 ` p2pnqr
n2 ` ¨ ¨ ¨ ` p`pnqr
n` ,
where pipnq are polynomials in n of degree mi ´ 1.
Example: an “ an´1 ` 5an´2 ´ an´3 ´ 8an´4 ´ 4an´5.Characteristic equation:
0 “ r5 ´ r4 ´ 5r3 ` r2 ` 8r ` 4 “ pr ` 1q3pr ´ 2q2.General solution:
an “ pα0 ` α1n` α2n2qp´1qn ` pβ0 ` β1nqp2q
n.
![Page 9: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/9.jpg)
Theorem: Solving linear homogeneous recurrencesLet c1, c2, . . . , ck be real numbers. Suppose that the characteristicequation has roots r1, r2, . . . , r` with multiplicities m1,m2, . . . ,m`.Then a sequence tanunPN is a solution of the recurrence relationan “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k if and only if
an “ p1pnqrn1 ` p2pnqr
n2 ` ¨ ¨ ¨ ` p`pnqr
n` ,
where pipnq are polynomials in n of degree mi ´ 1.
Example: an “ an´1 ` 5an´2 ´ an´3 ´ 8an´4 ´ 4an´5.
Characteristic equation:0 “ r5 ´ r4 ´ 5r3 ` r2 ` 8r ` 4 “ pr ` 1q3pr ´ 2q2.
General solution:an “ pα0 ` α1n` α2n
2qp´1qn ` pβ0 ` β1nqp2qn.
![Page 10: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/10.jpg)
Theorem: Solving linear homogeneous recurrencesLet c1, c2, . . . , ck be real numbers. Suppose that the characteristicequation has roots r1, r2, . . . , r` with multiplicities m1,m2, . . . ,m`.Then a sequence tanunPN is a solution of the recurrence relationan “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k if and only if
an “ p1pnqrn1 ` p2pnqr
n2 ` ¨ ¨ ¨ ` p`pnqr
n` ,
where pipnq are polynomials in n of degree mi ´ 1.
Example: an “ an´1 ` 5an´2 ´ an´3 ´ 8an´4 ´ 4an´5.Characteristic equation:
0 “ r5 ´ r4 ´ 5r3 ` r2 ` 8r ` 4
“ pr ` 1q3pr ´ 2q2.General solution:
an “ pα0 ` α1n` α2n2qp´1qn ` pβ0 ` β1nqp2q
n.
![Page 11: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/11.jpg)
Theorem: Solving linear homogeneous recurrencesLet c1, c2, . . . , ck be real numbers. Suppose that the characteristicequation has roots r1, r2, . . . , r` with multiplicities m1,m2, . . . ,m`.Then a sequence tanunPN is a solution of the recurrence relationan “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k if and only if
an “ p1pnqrn1 ` p2pnqr
n2 ` ¨ ¨ ¨ ` p`pnqr
n` ,
where pipnq are polynomials in n of degree mi ´ 1.
Example: an “ an´1 ` 5an´2 ´ an´3 ´ 8an´4 ´ 4an´5.Characteristic equation:
0 “ r5 ´ r4 ´ 5r3 ` r2 ` 8r ` 4 “ pr ` 1q3pr ´ 2q2.
General solution:an “ pα0 ` α1n` α2n
2qp´1qn ` pβ0 ` β1nqp2qn.
![Page 12: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/12.jpg)
Theorem: Solving linear homogeneous recurrencesLet c1, c2, . . . , ck be real numbers. Suppose that the characteristicequation has roots r1, r2, . . . , r` with multiplicities m1,m2, . . . ,m`.Then a sequence tanunPN is a solution of the recurrence relationan “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k if and only if
an “ p1pnqrn1 ` p2pnqr
n2 ` ¨ ¨ ¨ ` p`pnqr
n` ,
where pipnq are polynomials in n of degree mi ´ 1.
Example: an “ an´1 ` 5an´2 ´ an´3 ´ 8an´4 ´ 4an´5.Characteristic equation:
0 “ r5 ´ r4 ´ 5r3 ` r2 ` 8r ` 4 “ pr ` 1q3pr ´ 2q2.General solution:
an “ pα0 ` α1n` α2n2qp´1qn ` pβ0 ` β1nqp2q
n.
![Page 13: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/13.jpg)
Non-homogeneous equations
Suppose your recurrence is linear and constant coefficient in ai’s,but is not homogeneous. In other words, it is in the form
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k ` F pnq
where F pnq is a function only in n (no ai’s).
The relation
hn “ c1hn´1 ` c2hn´2 ` ¨ ¨ ¨ ` ckhn´k (so that an “ hn ` F pnq)
is called the associated homogeneous recurrence relation.
Ex: an “ 3an´1 ` 2n.F pnq“ 2n, Assoc. hom: hn “ 3hn´1.
Ex: an “ 5an´1 ´ 6an´2 ` 7n.F pnq“ 7n, Assoc. hom: hn “ 5hn´1 ´ 6hn´2.
The following theorem says that if we can find one solution to an,then the general solutions to hn will help us build all the rest ofthe solutions to an.
![Page 14: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/14.jpg)
Non-homogeneous equations
Suppose your recurrence is linear and constant coefficient in ai’s,but is not homogeneous. In other words, it is in the form
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k ` F pnq
where F pnq is a function only in n (no ai’s). The relation
hn “ c1hn´1 ` c2hn´2 ` ¨ ¨ ¨ ` ckhn´k (so that an “ hn ` F pnq)
is called the associated homogeneous recurrence relation.
Ex: an “ 3an´1 ` 2n.F pnq“ 2n, Assoc. hom: hn “ 3hn´1.
Ex: an “ 5an´1 ´ 6an´2 ` 7n.F pnq“ 7n, Assoc. hom: hn “ 5hn´1 ´ 6hn´2.
The following theorem says that if we can find one solution to an,then the general solutions to hn will help us build all the rest ofthe solutions to an.
![Page 15: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/15.jpg)
Non-homogeneous equations
Suppose your recurrence is linear and constant coefficient in ai’s,but is not homogeneous. In other words, it is in the form
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k ` F pnq
where F pnq is a function only in n (no ai’s). The relation
hn “ c1hn´1 ` c2hn´2 ` ¨ ¨ ¨ ` ckhn´k (so that an “ hn ` F pnq)
is called the associated homogeneous recurrence relation.
Ex: an “ 3an´1 ` 2n.
F pnq“ 2n, Assoc. hom: hn “ 3hn´1.
Ex: an “ 5an´1 ´ 6an´2 ` 7n.F pnq“ 7n, Assoc. hom: hn “ 5hn´1 ´ 6hn´2.
The following theorem says that if we can find one solution to an,then the general solutions to hn will help us build all the rest ofthe solutions to an.
![Page 16: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/16.jpg)
Non-homogeneous equations
Suppose your recurrence is linear and constant coefficient in ai’s,but is not homogeneous. In other words, it is in the form
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k ` F pnq
where F pnq is a function only in n (no ai’s). The relation
hn “ c1hn´1 ` c2hn´2 ` ¨ ¨ ¨ ` ckhn´k (so that an “ hn ` F pnq)
is called the associated homogeneous recurrence relation.
Ex: an “ 3an´1 ` 2n.F pnq“ 2n
, Assoc. hom: hn “ 3hn´1.
Ex: an “ 5an´1 ´ 6an´2 ` 7n.F pnq“ 7n, Assoc. hom: hn “ 5hn´1 ´ 6hn´2.
The following theorem says that if we can find one solution to an,then the general solutions to hn will help us build all the rest ofthe solutions to an.
![Page 17: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/17.jpg)
Non-homogeneous equations
Suppose your recurrence is linear and constant coefficient in ai’s,but is not homogeneous. In other words, it is in the form
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k ` F pnq
where F pnq is a function only in n (no ai’s). The relation
hn “ c1hn´1 ` c2hn´2 ` ¨ ¨ ¨ ` ckhn´k (so that an “ hn ` F pnq)
is called the associated homogeneous recurrence relation.
Ex: an “ 3an´1 ` 2n.F pnq“ 2n, Assoc. hom: hn “ 3hn´1.
Ex: an “ 5an´1 ´ 6an´2 ` 7n.F pnq“ 7n, Assoc. hom: hn “ 5hn´1 ´ 6hn´2.
The following theorem says that if we can find one solution to an,then the general solutions to hn will help us build all the rest ofthe solutions to an.
![Page 18: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/18.jpg)
Non-homogeneous equations
Suppose your recurrence is linear and constant coefficient in ai’s,but is not homogeneous. In other words, it is in the form
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k ` F pnq
where F pnq is a function only in n (no ai’s). The relation
hn “ c1hn´1 ` c2hn´2 ` ¨ ¨ ¨ ` ckhn´k (so that an “ hn ` F pnq)
is called the associated homogeneous recurrence relation.
Ex: an “ 3an´1 ` 2n.F pnq“ 2n, Assoc. hom: hn “ 3hn´1.
Ex: an “ 5an´1 ´ 6an´2 ` 7n.
F pnq“ 7n, Assoc. hom: hn “ 5hn´1 ´ 6hn´2.
The following theorem says that if we can find one solution to an,then the general solutions to hn will help us build all the rest ofthe solutions to an.
![Page 19: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/19.jpg)
Non-homogeneous equations
Suppose your recurrence is linear and constant coefficient in ai’s,but is not homogeneous. In other words, it is in the form
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k ` F pnq
where F pnq is a function only in n (no ai’s). The relation
hn “ c1hn´1 ` c2hn´2 ` ¨ ¨ ¨ ` ckhn´k (so that an “ hn ` F pnq)
is called the associated homogeneous recurrence relation.
Ex: an “ 3an´1 ` 2n.F pnq“ 2n, Assoc. hom: hn “ 3hn´1.
Ex: an “ 5an´1 ´ 6an´2 ` 7n.F pnq“ 7n
, Assoc. hom: hn “ 5hn´1 ´ 6hn´2.
The following theorem says that if we can find one solution to an,then the general solutions to hn will help us build all the rest ofthe solutions to an.
![Page 20: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/20.jpg)
Non-homogeneous equations
Suppose your recurrence is linear and constant coefficient in ai’s,but is not homogeneous. In other words, it is in the form
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k ` F pnq
where F pnq is a function only in n (no ai’s). The relation
hn “ c1hn´1 ` c2hn´2 ` ¨ ¨ ¨ ` ckhn´k (so that an “ hn ` F pnq)
is called the associated homogeneous recurrence relation.
Ex: an “ 3an´1 ` 2n.F pnq“ 2n, Assoc. hom: hn “ 3hn´1.
Ex: an “ 5an´1 ´ 6an´2 ` 7n.F pnq“ 7n, Assoc. hom: hn “ 5hn´1 ´ 6hn´2.
The following theorem says that if we can find one solution to an,then the general solutions to hn will help us build all the rest ofthe solutions to an.
![Page 21: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/21.jpg)
Non-homogeneous equations
Suppose your recurrence is linear and constant coefficient in ai’s,but is not homogeneous. In other words, it is in the form
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k ` F pnq
where F pnq is a function only in n (no ai’s). The relation
hn “ c1hn´1 ` c2hn´2 ` ¨ ¨ ¨ ` ckhn´k (so that an “ hn ` F pnq)
is called the associated homogeneous recurrence relation.
Ex: an “ 3an´1 ` 2n.F pnq“ 2n, Assoc. hom: hn “ 3hn´1.
Ex: an “ 5an´1 ´ 6an´2 ` 7n.F pnq“ 7n, Assoc. hom: hn “ 5hn´1 ´ 6hn´2.
The following theorem says that if we can find one solution to an,then the general solutions to hn will help us build all the rest ofthe solutions to an.
![Page 22: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/22.jpg)
Theorem: Solving non-homogeneous equations
(a) If tanunPN is one solution of the non-homogeneous linearrecurrence relation with constant coefficients
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k ` F pnq,then every solution is of the form tan “ an ` hnunPN, wherethnunPN is a solution of the associated homogeneousrecurrence relation.
(b) Finding an: If F pnq “ QpnqRn, where§ Qpnq is a polynomial in n, and§ R is a constant,
then there is a solution to an of the form
an “ nmqpnqRn
where§ degpqpnqq ď degpQpnqq, and§ m “ mult. of R in the characteristic equation (possibly 0).
Ex: Find all solutions to an “ 3an´1 ` 2n. What is the solutionwith a1 “ 3?
![Page 23: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/23.jpg)
Theorem: Solving non-homogeneous equations
(a) If tanunPN is one solution of the non-homogeneous linearrecurrence relation with constant coefficients
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k ` F pnq,then every solution is of the form tan “ an ` hnunPN, wherethnunPN is a solution of the associated homogeneousrecurrence relation.
(b) Finding an: If F pnq “ QpnqRn, where§ Qpnq is a polynomial in n, and§ R is a constant,
then there is a solution to an of the form
an “ nmqpnqRn
where§ degpqpnqq ď degpQpnqq, and§ m “ mult. of R in the characteristic equation (possibly 0).
Ex: Find all solutions to an “ 3an´1 ` 2n. What is the solutionwith a1 “ 3?
![Page 24: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/24.jpg)
Theorem: Solving non-homogeneous equations
(a) If tanunPN is one solution of the non-homogeneous linearrecurrence relation with constant coefficients
an “ c1an´1 ` c2an´2 ` ¨ ¨ ¨ ` ckan´k ` F pnq,then every solution is of the form tan “ an ` hnunPN, wherethnunPN is a solution of the associated homogeneousrecurrence relation.
(b) Finding an: If F pnq “ QpnqRn, where§ Qpnq is a polynomial in n, and§ R is a constant,
then there is a solution to an of the form
an “ nmqpnqRn
where§ degpqpnqq ď degpQpnqq, and§ m “ mult. of R in the characteristic equation (possibly 0).
Ex: Find all solutions to an “ 3an´1 ` 2n. What is the solutionwith a1 “ 3?
![Page 25: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/25.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ b4n (gives b “ 32{3)
General sol: an “ αp´2qn ` pα` βnq3n ` 323 4
n.
![Page 26: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/26.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ b4n (gives b “ 32{3)
General sol: an “ αp´2qn ` pα` βnq3n ` 323 4
n.
![Page 27: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/27.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ b4n (gives b “ 32{3)
General sol: an “ αp´2qn ` pα` βnq3n ` 323 4
n.
![Page 28: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/28.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18
“ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ b4n (gives b “ 32{3)
General sol: an “ αp´2qn ` pα` βnq3n ` 323 4
n.
![Page 29: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/29.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ b4n (gives b “ 32{3)
General sol: an “ αp´2qn ` pα` βnq3n ` 323 4
n.
![Page 30: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/30.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ b4n (gives b “ 32{3)
General sol: an “ αp´2qn ` pα` βnq3n ` 323 4
n.
![Page 31: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/31.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ b4n
(gives b “ 32{3)
General sol: an “ αp´2qn ` pα` βnq3n ` 323 4
n.
![Page 32: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/32.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ b4n (gives b “ 32{3)
General sol: an “ αp´2qn ` pα` βnq3n ` 323 4
n.
![Page 33: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/33.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ b4n (gives b “ 32{3)
General sol: an “ αp´2qn ` pα` βnq3n ` 323 4
n.
![Page 34: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/34.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ n4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nq4n
(gives b0 “ 1376{9 and b1 “ ´63{3)
General sol: an “ αp´2qn ` pα` βnq3n ` p13769 ´ 633 nq4
n.
![Page 35: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/35.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ n4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nq4n
(gives b0 “ 1376{9 and b1 “ ´63{3)
General sol: an “ αp´2qn ` pα` βnq3n ` p13769 ´ 633 nq4
n.
![Page 36: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/36.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ n4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18
“ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nq4n
(gives b0 “ 1376{9 and b1 “ ´63{3)
General sol: an “ αp´2qn ` pα` βnq3n ` p13769 ´ 633 nq4
n.
![Page 37: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/37.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ n4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nq4n
(gives b0 “ 1376{9 and b1 “ ´63{3)
General sol: an “ αp´2qn ` pα` βnq3n ` p13769 ´ 633 nq4
n.
![Page 38: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/38.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ n4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nq4n
(gives b0 “ 1376{9 and b1 “ ´63{3)
General sol: an “ αp´2qn ` pα` βnq3n ` p13769 ´ 633 nq4
n.
![Page 39: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/39.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ n4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nq4n
(gives b0 “ 1376{9 and b1 “ ´63{3)
General sol: an “ αp´2qn ` pα` βnq3n ` p13769 ´ 633 nq4
n.
![Page 40: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/40.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ n4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nq4n
(gives b0 “ 1376{9 and b1 “ ´63{3)
General sol: an “ αp´2qn ` pα` βnq3n ` p13769 ´ 633 nq4
n.
![Page 41: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/41.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n4n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ n4n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nq4n
(gives b0 “ 1376{9 and b1 “ ´63{3)
General sol: an “ αp´2qn ` pα` βnq3n ` p13769 ´ 633 nq4
n.
![Page 42: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/42.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ bn23n
(gives b “ 3{2)
General sol: an “ αp´2qn ` pα` βnq3n ` 32n
23n.
![Page 43: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/43.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ bn23n
(gives b “ 3{2)
General sol: an “ αp´2qn ` pα` βnq3n ` 32n
23n.
![Page 44: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/44.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18
“ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ bn23n
(gives b “ 3{2)
General sol: an “ αp´2qn ` pα` βnq3n ` 32n
23n.
![Page 45: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/45.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ bn23n
(gives b “ 3{2)
General sol: an “ αp´2qn ` pα` βnq3n ` 32n
23n.
![Page 46: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/46.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ bn23n
(gives b “ 3{2)
General sol: an “ αp´2qn ` pα` βnq3n ` 32n
23n.
![Page 47: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/47.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ bn23n
(gives b “ 3{2)
General sol: an “ αp´2qn ` pα` βnq3n ` 32n
23n.
![Page 48: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/48.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ bn23n
(gives b “ 3{2)
General sol: an “ αp´2qn ` pα` βnq3n ` 32n
23n.
![Page 49: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/49.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` 3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ bn23n
(gives b “ 3{2)
General sol: an “ αp´2qn ` pα` βnq3n ` 32n
23n.
![Page 50: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/50.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nqn23n
(gives b0 “ 21{50 and b1 “ 1{10)
General sol: an “ αp´2qn ` pα` βnq3n ` p2150 `110nqn
23n.
![Page 51: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/51.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nqn23n
(gives b0 “ 21{50 and b1 “ 1{10)
General sol: an “ αp´2qn ` pα` βnq3n ` p2150 `110nqn
23n.
![Page 52: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/52.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18
“ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nqn23n
(gives b0 “ 21{50 and b1 “ 1{10)
General sol: an “ αp´2qn ` pα` βnq3n ` p2150 `110nqn
23n.
![Page 53: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/53.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nqn23n
(gives b0 “ 21{50 and b1 “ 1{10)
General sol: an “ αp´2qn ` pα` βnq3n ` p2150 `110nqn
23n.
![Page 54: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/54.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nqn23n
(gives b0 “ 21{50 and b1 “ 1{10)
General sol: an “ αp´2qn ` pα` βnq3n ` p2150 `110nqn
23n.
![Page 55: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/55.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nqn23n
(gives b0 “ 21{50 and b1 “ 1{10)
General sol: an “ αp´2qn ` pα` βnq3n ` p2150 `110nqn
23n.
![Page 56: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/56.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nqn23n
(gives b0 “ 21{50 and b1 “ 1{10)
General sol: an “ αp´2qn ` pα` βnq3n ` p2150 `110nqn
23n.
![Page 57: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/57.jpg)
1. Break the sequence into two parts: homogeneous hn and a
function of n: an “ hn ` F pnq .
2. Solve for hn:§ compute the characteristic equation;§ factor to compute roots and multiplicities;§ build the general solution to hn.
3. Fine one solution an by guessing something of a similar form.
If F pnq “ QpnqRn, guess an “ nmqpnqRn
where m “ mult of R, and qn “ b0 ` b1n` b2n2 ` ¨ ¨ ¨ ` bdn
d
where d “ degpQpnqq.
Example: an “ 4an´1 ` 3an´2 ´ 18an´3 ` n3n
Homog: hn “ 4hn´1 ` 3hn´2 ´ 18hn´3 and F pnq “ 3n.
Char eq: 0 “ r3 ´ 4r2 ´ 3r ` 18 “ pr ` 2qpr ´ 3q2
Homog sol: hn “ αp´2qn ` pα` βnq3n
Particular solution guess: an “ pb0 ` b1nqn23n
(gives b0 “ 21{50 and b1 “ 1{10)
General sol: an “ αp´2qn ` pα` βnq3n ` p2150 `110nqn
23n.
![Page 58: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/58.jpg)
Section 8.4: Generating functions.
Taylor series to know and love:
p1` xqn “nÿ
k“0
ˆ
n
k
˙
xk “ 1`
ˆ
n
1
˙
x`
ˆ
n
2
˙
x2 ` ¨ ¨ ¨ ` xn (finite)
1´ xn
1´ x“
n´1ÿ
k“0
xk “ 1` x` x2 ` ¨ ¨ ¨ ` xn´1 (finite)
1
1´ x“
8ÿ
k“0
xk “ 1` x` x2 ` ¨ ¨ ¨ (infinite)
ex “8ÿ
k“0
xk{k! “ 1` x`x2
2`x3
3!` ¨ ¨ ¨ (infinite)
Combining series: Let fpxq “8ÿ
k“0
akxk and gpxq “
8ÿ
k“0
bkxk. Then
fpxq`gpxq “8ÿ
k“0
pak`bkqxk and fpxqgpxq “
8ÿ
k“0
˜
kÿ
i“0
aibk´i
¸
xk.
![Page 59: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/59.jpg)
Section 8.4: Generating functions.
Taylor series to know and love:
p1` xqn “nÿ
k“0
ˆ
n
k
˙
xk “ 1`
ˆ
n
1
˙
x`
ˆ
n
2
˙
x2 ` ¨ ¨ ¨ ` xn (finite)
1´ xn
1´ x“
n´1ÿ
k“0
xk “ 1` x` x2 ` ¨ ¨ ¨ ` xn´1 (finite)
1
1´ x“
8ÿ
k“0
xk “ 1` x` x2 ` ¨ ¨ ¨ (infinite)
ex “8ÿ
k“0
xk{k! “ 1` x`x2
2`x3
3!` ¨ ¨ ¨ (infinite)
Combining series: Let fpxq “8ÿ
k“0
akxk and gpxq “
8ÿ
k“0
bkxk. Then
fpxq`gpxq “8ÿ
k“0
pak`bkqxk and fpxqgpxq “
8ÿ
k“0
˜
kÿ
i“0
aibk´i
¸
xk.
![Page 60: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/60.jpg)
Section 8.4: Generating functions.
Taylor series to know and love:
p1` xqn “nÿ
k“0
ˆ
n
k
˙
xk “ 1`
ˆ
n
1
˙
x`
ˆ
n
2
˙
x2 ` ¨ ¨ ¨ ` xn (finite)
1´ xn
1´ x“
n´1ÿ
k“0
xk “ 1` x` x2 ` ¨ ¨ ¨ ` xn´1 (finite)
1
1´ x“
8ÿ
k“0
xk “ 1` x` x2 ` ¨ ¨ ¨ (infinite)
ex “8ÿ
k“0
xk{k! “ 1` x`x2
2`x3
3!` ¨ ¨ ¨ (infinite)
Combining series: Let fpxq “8ÿ
k“0
akxk and gpxq “
8ÿ
k“0
bkxk. Then
fpxq`gpxq “8ÿ
k“0
pak`bkqxk and fpxqgpxq “
8ÿ
k“0
˜
kÿ
i“0
aibk´i
¸
xk.
![Page 61: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/61.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 1: Use the multiplication rule,
˜
8ÿ
k“0
akxk
¸˜
8ÿ
k“0
bkxk
¸
“
8ÿ
k“0
˜
kÿ
i“0
aibk´i
¸
xk,
on1
p1´ xq2“
1
1´ x˚
1
1´ x“
˜
8ÿ
k“0
xk
¸˜
8ÿ
k“0
xk
¸
.
Here, ai “ bi “ 1 for all i.So
kÿ
i“0
aibk´i “kÿ
i“0
1 ˚ 1 “ k ` 1.
Thus
1
p1´ xq2“
8ÿ
k“0
pk ` 1qxk “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
![Page 62: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/62.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 1: Use the multiplication rule,
˜
8ÿ
k“0
akxk
¸˜
8ÿ
k“0
bkxk
¸
“
8ÿ
k“0
˜
kÿ
i“0
aibk´i
¸
xk,
on1
p1´ xq2“
1
1´ x˚
1
1´ x“
˜
8ÿ
k“0
xk
¸˜
8ÿ
k“0
xk
¸
.
Here, ai “ bi “ 1 for all i.So
kÿ
i“0
aibk´i “kÿ
i“0
1 ˚ 1 “ k ` 1.
Thus
1
p1´ xq2“
8ÿ
k“0
pk ` 1qxk “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
![Page 63: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/63.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 1: Use the multiplication rule,
˜
8ÿ
k“0
akxk
¸˜
8ÿ
k“0
bkxk
¸
“
8ÿ
k“0
˜
kÿ
i“0
aibk´i
¸
xk,
on1
p1´ xq2“
1
1´ x˚
1
1´ x“
˜
8ÿ
k“0
xk
¸˜
8ÿ
k“0
xk
¸
.
Here, ai “ bi “ 1 for all i.
Sokÿ
i“0
aibk´i “kÿ
i“0
1 ˚ 1 “ k ` 1.
Thus
1
p1´ xq2“
8ÿ
k“0
pk ` 1qxk “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
![Page 64: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/64.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 1: Use the multiplication rule,
˜
8ÿ
k“0
akxk
¸˜
8ÿ
k“0
bkxk
¸
“
8ÿ
k“0
˜
kÿ
i“0
aibk´i
¸
xk,
on1
p1´ xq2“
1
1´ x˚
1
1´ x“
˜
8ÿ
k“0
xk
¸˜
8ÿ
k“0
xk
¸
.
Here, ai “ bi “ 1 for all i.So
kÿ
i“0
aibk´i
“
kÿ
i“0
1 ˚ 1 “ k ` 1.
Thus
1
p1´ xq2“
8ÿ
k“0
pk ` 1qxk “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
![Page 65: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/65.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 1: Use the multiplication rule,
˜
8ÿ
k“0
akxk
¸˜
8ÿ
k“0
bkxk
¸
“
8ÿ
k“0
˜
kÿ
i“0
aibk´i
¸
xk,
on1
p1´ xq2“
1
1´ x˚
1
1´ x“
˜
8ÿ
k“0
xk
¸˜
8ÿ
k“0
xk
¸
.
Here, ai “ bi “ 1 for all i.So
kÿ
i“0
aibk´i “kÿ
i“0
1 ˚ 1
“ k ` 1.
Thus
1
p1´ xq2“
8ÿ
k“0
pk ` 1qxk “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
![Page 66: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/66.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 1: Use the multiplication rule,
˜
8ÿ
k“0
akxk
¸˜
8ÿ
k“0
bkxk
¸
“
8ÿ
k“0
˜
kÿ
i“0
aibk´i
¸
xk,
on1
p1´ xq2“
1
1´ x˚
1
1´ x“
˜
8ÿ
k“0
xk
¸˜
8ÿ
k“0
xk
¸
.
Here, ai “ bi “ 1 for all i.So
kÿ
i“0
aibk´i “kÿ
i“0
1 ˚ 1 “ k ` 1.
Thus
1
p1´ xq2“
8ÿ
k“0
pk ` 1qxk “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
![Page 67: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/67.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 1: Use the multiplication rule,
˜
8ÿ
k“0
akxk
¸˜
8ÿ
k“0
bkxk
¸
“
8ÿ
k“0
˜
kÿ
i“0
aibk´i
¸
xk,
on1
p1´ xq2“
1
1´ x˚
1
1´ x“
˜
8ÿ
k“0
xk
¸˜
8ÿ
k“0
xk
¸
.
Here, ai “ bi “ 1 for all i.So
kÿ
i“0
aibk´i “kÿ
i“0
1 ˚ 1 “ k ` 1.
Thus
1
p1´ xq2“
8ÿ
k“0
pk ` 1qxk
“ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
![Page 68: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/68.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 1: Use the multiplication rule,
˜
8ÿ
k“0
akxk
¸˜
8ÿ
k“0
bkxk
¸
“
8ÿ
k“0
˜
kÿ
i“0
aibk´i
¸
xk,
on1
p1´ xq2“
1
1´ x˚
1
1´ x“
˜
8ÿ
k“0
xk
¸˜
8ÿ
k“0
xk
¸
.
Here, ai “ bi “ 1 for all i.So
kÿ
i“0
aibk´i “kÿ
i“0
1 ˚ 1 “ k ` 1.
Thus
1
p1´ xq2“
8ÿ
k“0
pk ` 1qxk “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
![Page 69: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/69.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 2: Use derivatives, noting that
d
dx
1
1´ x“
d
dxp1´ xq´1 “ p´1qp1´ xq´2p´1q “
1
p1´ xq2.
Thus,
1
p1´ xq2“
d
dxp1´ xq´1
“d
dx
8ÿ
k“0
xk
“
8ÿ
k“0
d
dxxk
“
8ÿ
k“0
kxk´1 (change summation: let j “ k ´ 1)
“
8ÿ
j“0
pj ` 1qxj “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
You try: Exercise 36
![Page 70: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/70.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 2: Use derivatives, noting that
d
dx
1
1´ x“
d
dxp1´ xq´1
“ p´1qp1´ xq´2p´1q “1
p1´ xq2.
Thus,
1
p1´ xq2“
d
dxp1´ xq´1
“d
dx
8ÿ
k“0
xk
“
8ÿ
k“0
d
dxxk
“
8ÿ
k“0
kxk´1 (change summation: let j “ k ´ 1)
“
8ÿ
j“0
pj ` 1qxj “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
You try: Exercise 36
![Page 71: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/71.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 2: Use derivatives, noting that
d
dx
1
1´ x“
d
dxp1´ xq´1 “ p´1qp1´ xq´2p´1q
“1
p1´ xq2.
Thus,
1
p1´ xq2“
d
dxp1´ xq´1
“d
dx
8ÿ
k“0
xk
“
8ÿ
k“0
d
dxxk
“
8ÿ
k“0
kxk´1 (change summation: let j “ k ´ 1)
“
8ÿ
j“0
pj ` 1qxj “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
You try: Exercise 36
![Page 72: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/72.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 2: Use derivatives, noting that
d
dx
1
1´ x“
d
dxp1´ xq´1 “ p´1qp1´ xq´2p´1q “
1
p1´ xq2.
Thus,
1
p1´ xq2“
d
dxp1´ xq´1
“d
dx
8ÿ
k“0
xk
“
8ÿ
k“0
d
dxxk
“
8ÿ
k“0
kxk´1 (change summation: let j “ k ´ 1)
“
8ÿ
j“0
pj ` 1qxj “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
You try: Exercise 36
![Page 73: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/73.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 2: Use derivatives, noting that
d
dx
1
1´ x“
d
dxp1´ xq´1 “ p´1qp1´ xq´2p´1q “
1
p1´ xq2.
Thus,
1
p1´ xq2“
d
dxp1´ xq´1 “
d
dx
8ÿ
k“0
xk
“
8ÿ
k“0
d
dxxk
“
8ÿ
k“0
kxk´1 (change summation: let j “ k ´ 1)
“
8ÿ
j“0
pj ` 1qxj “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
You try: Exercise 36
![Page 74: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/74.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 2: Use derivatives, noting that
d
dx
1
1´ x“
d
dxp1´ xq´1 “ p´1qp1´ xq´2p´1q “
1
p1´ xq2.
Thus,
1
p1´ xq2“
d
dxp1´ xq´1 “
d
dx
8ÿ
k“0
xk
“
8ÿ
k“0
d
dxxk
“
8ÿ
k“0
kxk´1 (change summation: let j “ k ´ 1)
“
8ÿ
j“0
pj ` 1qxj “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
You try: Exercise 36
![Page 75: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/75.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 2: Use derivatives, noting that
d
dx
1
1´ x“
d
dxp1´ xq´1 “ p´1qp1´ xq´2p´1q “
1
p1´ xq2.
Thus,
1
p1´ xq2“
d
dxp1´ xq´1 “
d
dx
8ÿ
k“0
xk
“
8ÿ
k“0
d
dxxk
“
8ÿ
k“0
kxk´1
(change summation: let j “ k ´ 1)
“
8ÿ
j“0
pj ` 1qxj “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
You try: Exercise 36
![Page 76: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/76.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 2: Use derivatives, noting that
d
dx
1
1´ x“
d
dxp1´ xq´1 “ p´1qp1´ xq´2p´1q “
1
p1´ xq2.
Thus,
1
p1´ xq2“
d
dxp1´ xq´1 “
d
dx
8ÿ
k“0
xk
“
8ÿ
k“0
d
dxxk
“
8ÿ
k“0
kxk´1 (change summation: let j “ k ´ 1)
“
8ÿ
j“0
pj ` 1qxj “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
You try: Exercise 36
![Page 77: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/77.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 2: Use derivatives, noting that
d
dx
1
1´ x“
d
dxp1´ xq´1 “ p´1qp1´ xq´2p´1q “
1
p1´ xq2.
Thus,
1
p1´ xq2“
d
dxp1´ xq´1 “
d
dx
8ÿ
k“0
xk
“
8ÿ
k“0
d
dxxk
“
8ÿ
k“0
kxk´1 (change summation: let j “ k ´ 1)
“
8ÿ
j“0
pj ` 1qxj
“ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
You try: Exercise 36
![Page 78: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/78.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 2: Use derivatives, noting that
d
dx
1
1´ x“
d
dxp1´ xq´1 “ p´1qp1´ xq´2p´1q “
1
p1´ xq2.
Thus,
1
p1´ xq2“
d
dxp1´ xq´1 “
d
dx
8ÿ
k“0
xk
“
8ÿ
k“0
d
dxxk
“
8ÿ
k“0
kxk´1 (change summation: let j “ k ´ 1)
“
8ÿ
j“0
pj ` 1qxj “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
You try: Exercise 36
![Page 79: Today: Section 8.2, continued. To solve recurrence ... · To solve recurrence relations, the best we can do is make educated guesses, according to what kind of relation we have. Alinear](https://reader033.vdocuments.site/reader033/viewer/2022042918/5f5e417070718e02e97e7e97/html5/thumbnails/79.jpg)
Example: Compute the series for 1p1´xq2
using 11´x “
ř8k“0 x
k.
Approach 2: Use derivatives, noting that
d
dx
1
1´ x“
d
dxp1´ xq´1 “ p´1qp1´ xq´2p´1q “
1
p1´ xq2.
Thus,
1
p1´ xq2“
d
dxp1´ xq´1 “
d
dx
8ÿ
k“0
xk
“
8ÿ
k“0
d
dxxk
“
8ÿ
k“0
kxk´1 (change summation: let j “ k ´ 1)
“
8ÿ
j“0
pj ` 1qxj “ 1` 2x` 3x2 ` 4x3 ` ¨ ¨ ¨ .
You try: Exercise 36