Algebra II Unit 1 Lesson 2, 3 & 5

Lesson 2: F 502. Find the next term in a sequence described recursively.

Lesson 3: F 603. Find a recursive expression for the general term in a sequence described recursively.

Lesson 5: Exhibit knowledge of geometric sequences.

Lesson 4: Solve complex arithmetic problems involving percent of increase /decrease or requiring integration of several concepts

(i.e. using several ratios, comparing percentages or averages).

Do Now:

Identify each sequence as arithmetic, geometric or neither.

a.14, 7, 3.5, 1.75,…

b.47, 41, 35, 29,…

c.1, 1, 2, 3, 5, 8,…d.The non-horizontal cards

According to the US EPA, each American produced an average of 978 lb of trash in 1960.This increased to 1336 lb in 1980. By 2000, trash production had risen to 1646 lb/yr per person. How would you find the total amount of trash a person produced in a lifetime?

Arithmetic Series

Key Concepts:

We are learning about mathematical series and the summation notation used to represent them.

Write recursive and explicit formulas for the terms of the series and find partial sums of the series.

Vocabulary

Summation SymbolWhen we don’t want to write out a whole bunch of numbers in the series, the summation symbol is used when writing a series. The limits are the greatest and least values of n.

Summation symbol

Upper Limit (greatest value of n)

Lower Limit (least value of n)

Explicit function for the sequence

So, the way this works is plug in n=1 to the equation and continue through n=3.

(5*1 + 1) + (5*2 +1) + (5*3 + 1) = 33

Understanding Sigma Notation

Evaluate the following expressions

4

1

3n

n

1

2

)14(p

p

5

1

2 )2(k

k

What did Gauss do?

ExamplesEx. 1.) Find the sum of the first 50 multiples of 6:

Ex.2.) Find the sum of the first 75 even numbers starting with 2.

50,...18,12,6 u

Find the indicated values

1.

2. if

3. if

18

4

16k

k

9S 65 nun

18u 43 nun

The concert hall in the picture below has 59 seats in Row 1, 63 seats in Row 2, 67 seats in Row 3, and so on.

How many seats are there in all 35 rows of the concert hall?

Suppose that Seat 1 is at the left end of Row 1 and that Seat 60 is at the left end of Row 2. Describe the location of Seat 970.

Practice

Page: 633 #(1 – 5)

Sum of a Finite Arithmetic Series

)(2

1 nn aan

S

119)295(2

7nS

Let’s try one: evaluate the series:

5, 9, 13,17,21,25,29

Writing the given series in summation form.

Evaluate: Yes, you can add manually. But let’s try using the shortcut:

)(2

1 nn aan

S

642)112102(2

6nS

Practice

Find the number of terms, the first term and the last term. Then evaluate the series:

Ex.1 Ex.2

Notice we can use the shortcut here:

Why is the answer not 58?

Note: this is NOT an arithmetic series. You can NOT use the shortcut; you have to manually calculate all values.

25)5(5)72(2

10nS

Arithmetic Sequence:Arithmetic Sequence:

The difference between consecutive The difference between consecutive terms is constant (or the same).terms is constant (or the same).

The constant difference is also known as The constant difference is also known as the the common difference common difference (d).(d).

(It’s also the number you are adding (It’s also the number you are adding every time!)every time!)

The general form of an ARITHMETIC sequence.

1aFirst Term:

Second Term: 2 1a a d

Third Term:

Fourth Term:

Fifth Term:

3 1 2a a d

4 1 3a a d

5 1 4a a d

nth Term: 1 1na a n d

Formula for the nth term of an ARITHMETIC sequence.

1 1na a n d

The nth termna

The term numbern

The common differenced

1 The 1st terma

If we know any

If we know any three of these we

three of these we ought to be able

ought to be able to find the fourth.

to find the fourth.

Example: Decide whether each Example: Decide whether each sequence is arithmetic.sequence is arithmetic.

-10,-6,-2,0,2,6,10,…-10,-6,-2,0,2,6,10,…

-6-(-10)=4-6-(-10)=4

-2-(-6)=4-2-(-6)=4

0-(-2)=20-(-2)=2

2-0=22-0=2

6-2=46-2=4

10-6=410-6=4

Not arithmetic (because Not arithmetic (because the differences are not the differences are not the same)the same)

5,11,17,23,29,…5,11,17,23,29,…

11-5=611-5=6

17-11=617-11=6

23-17=623-17=6

29-23=629-23=6

Arithmetic (common Arithmetic (common difference is 6)difference is 6)

Rule for an Arithmetic Rule for an Arithmetic SequenceSequence

n = number of termsn = number of terms

aann = last term = last term

aann= a= a11+(n-1)d+(n-1)d

ExampleExample: Write a rule for the nth : Write a rule for the nth term of the sequence 32,47,62,77,… . term of the sequence 32,47,62,77,… . Then, find u Then, find u1212..

There is a common difference where d=15, There is a common difference where d=15, therefore the sequence is arithmetic.therefore the sequence is arithmetic.

Use uUse unn=u=u11+u(n-1)d+u(n-1)d

uunn=32+(n-1)(15) =32+(n-1)(15)

uunn=32+15n-15=32+15n-15

uunn=17+15n=17+15n

uu1212=17+15(12)=197=17+15(12)=197

ExampleExample: One term of an arithmetic sequence : One term of an arithmetic sequence is ais a88=50. The common difference is 0.25. =50. The common difference is 0.25.

Write a rule for the nth term.Write a rule for the nth term.Use aUse ann=a=a11+(n-1)d to find the 1+(n-1)d to find the 1stst term! term!

aa88=a=a11+(8-1)(.25)+(8-1)(.25)

50=a50=a11+(7)(.25)+(7)(.25)

50=a50=a11+1.75+1.75

48.25=a48.25=a11

* Now, use a* Now, use ann=a=a11+(n-1)d to find the rule.+(n-1)d to find the rule.

aann=48.25+(n-1)(.25)=48.25+(n-1)(.25)

aann=48.25+.25n-.25=48.25+.25n-.25

aann=48+.25n=48+.25n

Example: Two terms of an arithmetic sequence are Example: Two terms of an arithmetic sequence are aa55=10 and a=10 and a3030=110. Write a rule for the nth term.=110. Write a rule for the nth term.

Begin by writing 2 equations; one for each term given.Begin by writing 2 equations; one for each term given.

aa55=a=a11+(5-1)d OR 10=a+(5-1)d OR 10=a11+4d+4d

AndAnd

aa3030=a=a11+(30-1)d OR 110=a+(30-1)d OR 110=a11+29d+29d

Now use the 2 equations to solve for aNow use the 2 equations to solve for a11 & d. & d.

10=a10=a11+4d+4d

110=a110=a11+29d (subtract the equations to cancel a+29d (subtract the equations to cancel a11))

-100= -25d -100= -25d

So, d=4 and aSo, d=4 and a11=-6 (now find the rule)=-6 (now find the rule)

aann=a=a11+(n-1)d+(n-1)d

aann=-6+(n-1)(4) OR a=-6+(n-1)(4) OR ann=-10+4n=-10+4n

Arithmetic SeriesArithmetic Series

The sum of the terms The sum of the terms in an arithmetic in an arithmetic sequencesequence

The formula to find the The formula to find the sum of a finite sum of a finite arithmetic series is:arithmetic series is:

2

1 nn

aanS

# of terms# of terms

11stst Term Term

Last Last TermTerm

ExampleExample: Consider the arithmetic : Consider the arithmetic series 20+18+16+14+… .series 20+18+16+14+… .

Find n such that SFind n such that Snn=-760=-760

2

1 nn

aanS

2

28202525S 100)4(2525 S

2

1 nn

aanS

2

)222(20760

nn

--1520=n(20+22-2n)1520=n(20+22-2n)

-1520=-2n-1520=-2n22+42n+42n

2n2n22-42n-1520=0-42n-1520=0

nn22-21n-760=0-21n-760=0

(n-40)(n+19)=0(n-40)(n+19)=0

n=40 or n=-19n=40 or n=-19

Always choose the positive solution!Always choose the positive solution!

2

)222(20760

nn

An introduction…………Sequence Sum Sequence Sum

1, 4, 7,10,13

9,1, 7, 15

6.2, 6.6, 7, 7.4

, 3, 6

Arithmetic Sequences

ADDTo get next term

2, 4, 8,16, 32

9, 3,1, 1/ 3

1,1/ 4,1/16,1/ 64

, 2.5 , 6.25

Geometric Sequences

MULTIPLYTo get next term

35

12

27.2

3 9

62

20 / 3

85 / 64

9.75

Vocabulary of Sequences (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

n 1

n 1 n

nth term of arithmetic sequence

sum of n terms of arithmetic sequen

a a n 1 d

nS a a

2ce

Example: The nth Partial Sum

The sum of the first n terms of an infinite sequence is called the nth partial sum.

1( )2n nnS a a

Vocabulary of Sequences (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

n 1

n 1

n1

n

nth term of geometric sequence

sum of n terms of geometric sequ

a a r

a r 1S

r 1ence

1, 4, 7, 10, 13, …. Infinite Arithmetic No Sum

3, 7, 11, …, 51 Finite Arithmetic n 1 n

nS a a

2

1, 2, 4, …, 64 Finite Geometric n

1

n

a r 1S

r 1

1, 2, 4, 8, … Infinite Geometricr > 1r < -1

No Sum

1 1 13,1, , , ...

3 9 27Infinite Geometric

-1 < r < 11a

S1 r

Find the sum, if possible: 1 1 11 ...

2 4 8

1 112 4r

11 22

1 r 1 Yes

1a 1S 2

11 r 12

Find the sum, if possible: 2 2 8 16 2 ...

8 16 2r 2 2

82 2 1 r 1 No

NO SUM

Find the sum, if possible: 2 1 1 1...

3 3 6 12

1 113 6r

2 1 23 3

1 r 1 Yes

1

2a 43S

11 r 312

Find the sum, if possible: 2 4 8

...7 7 7

4 87 7r 22 47 7

1 r 1 No

NO SUM

Find the sum, if possible: 510 5 ...

2

55 12r

10 5 2 1 r 1 Yes

1a 10S 20

11 r 12

The Bouncing Ball Problem – Version A

A ball is dropped from a height of 50 feet. It rebounds 4/5 of

it’s height, and continues this pattern until it stops. How far

does the ball travel?50

40

32

32/5

40

32

32/5

40S 45

504

10

1554

The Bouncing Ball Problem – Version B

A ball is thrown 100 feet into the air. It rebounds 3/4 of

it’s height, and continues this pattern until it stops. How far

does the ball travel?

100

75

225/4

100

75

225/4

10S 80

100

4 43

1

0

10

3

Sigma Notation

B

nn A

a

UPPER BOUND(NUMBER)

LOWER BOUND(NUMBER)

SIGMA(SUM OF TERMS) NTH TERM

(SEQUENCE)

j

4

1

j 2

21 2 2 3 2 24 18

7

4a

2a 42 2 5 2 6 72 44

n

n 0

4

0.5 2

00.5 2 10.5 2 20.5 2 30.5 2 40.5 2

33.5

0

n

b

36

5

0

36

5

13

65

23

65

...

1aS

1 r

6

153

15

2

x

3

7

2x 1

2 1 2 8 1 2 9 1 ...7 2 123

n 1 n

2n 1S a a 15

2

3

2

747

527

1

b

9

4

4b 3

4 3 4 5 3 4 6 3 ...4 4 319

n 1 n

1n 1S a a 19

2

9

2

479

784

Rewrite using sigma notation: 3 + 6 + 9 + 12

Arithmetic, d= 3

n 1a a n 1 d

na 3 n 1 3

na 3n4

1n

3n

Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1

Geometric, r = ½

n 1n 1a a r

n 1

n

1a 16

2

n 1

n

5

1

116

2

Rewrite using sigma notation: 19 + 18 + 16 + 12 + 4

Not Arithmetic, Not Geometric

n 1na 20 2

n 1

n

5

1

20 2

19 + 18 + 16 + 12 + 4 -1 -2 -4 -8

Rewrite the following using sigma notation:3 9 27

...5 10 15

Numerator is geometric, r = 3Denominator is arithmetic d= 5

NUMERATOR: n 1

n3 9 27 ... a 3 3

DENOMINATOR: n n5 10 15 ... a 5 n 1 5 a 5n

SIGMA NOTATION: 1

1

n

n 5n

3 3