Sequences and Series

On occasion, it is convenient to begin subscripting a sequence with 0 instead of 1 so that the terms of the sequence become

0 1 2 3, , , ,...a a a a

Example 1.

Write the first four terms of the sequences given by

a. 3 2 b. 3 1n

n na n a

a. 1 3 1 2 1a

2 3 2 2 4a

3 3 3 2 7a 4 3 4 2 10a

b. 1

1 3 1 2a

2

2 3 1 4a 3

3 3 1 2a

4

4 3 1 4a

Example 2.

1Write the first five terms of the sequence given by

2 1

n

na n

1

1

1 11

2 1 1 1a

2

2

1 1

2 2 1 3a

3

3

1 1

2 3 1 5a

4

4

1 1

2 4 1 7a

5

5

1 1

2 5 1 9a

Simply listing the first few terms is not sufficient to define a unique sequence-----the nth term must be given.

1 1 1 1 1, , , ,... ,...

2 4 8 16 2n 2

1 1 1 1 6, , , ,... ,...

2 4 8 16 1 6n n n

Although the first three terms are the same, these are different sequences

We can only write an apparent nth term.

There may be others

Example 3.

Write an expression for the apparent th term of each sequence.

a. 1, 3, 5, 7,... b. 2, 5, 10, 17,...nn a

:1 2 3 4 ...n n

:1 3 5 7 ... nterms a

apparent pattern: 2 1na n

:1 2 3 4 ...n n

: 2 5 10 17 ... nterms a

apparent pattern:2 1na n

Example 4.

Write an expression for the apparent th term of the sequence:

2 3 4 5, , , ,...

1 2 3 4

nn a

1n

na

n

Some sequences are defined recursively.

To define a sequence recursively, you need to be given one or more of the first few terms.

0 1 2 3 1 2

Example 5.

The Fibonacci sequence is defined recursively as follows:

1 1 2 3 2k k ka a a a a a a where k

Write the first five terms of this sequence.

0 1 2 31 1 2 3a a a a

4 3 2 3 2 5a a a 5 4 3 5 3 8a a a

The subscripts of the sequence make up the domain of the sequence and they identify the location of a term within the sequence.

Factorial Notation

If is a positive integer, factorial is defined as

! 1 2 3 4 ... 1

n n

n n n

zero factorial is defined as 0! = 1

Factorials follow the same rules for order of operations as exponents.

2n! = 2(n!) = 2 1 2 3 4 ... n

2 ! 1 2 3 4 ... 2n n

Example 6.

2Write the first five terms of the sequence given by .

!Begin with 0

n

na nn

0

0

2 11

0! 1a

1

1

2 22

1! 1a

2

2

2 42

2! 2a

3

3

2 8 4

3! 6 3a

4

4

2 16 2

4! 24 3a

Example 7.

Evaluate each factorial expression.

8! 2!6! !a. b. c.

2!6! 3!5! 1 !

2 2 ! 2 1 !2 !d. e. f.

2 4 ! ! 2 !

n

n

n nn

n n n

8! 8 7 6! 8 7 56a. 28

2!6! 2!6! 2 2

2!6 5! 6

b. 23 2!5! 3

1 2 3 ... 1!c.

1 ! 1 2 3 ... 1

n nnn

n n

2

2 2 ! 1 2 3 2 2 1 1d.

2 4 ! 1 2 3 2 2 2 3 2 4 2 3 2 4 4 14 12

n n

n n n n n n n n

2 !e. =2

!

n

n 2 1 ! 1 2 3 2 2 1

f. 2 12 ! 1 2 3 2

n n nn

n n

There is a convenient notation for the sum of the terms of a finite sequence. It is called summation notation or sigma notation because it involves the use of the uppercase Greek letter sigma.

5 6 82

1 3 0

Example 8.

1a. 3 b. (1 ) c.

!i k n

i kn

5

1

a. 3 3 1 3 2 3 3 3 4 3 5i

i

3 1 2 3 4 5 3 15 45

6

2 2 2 2 2

3

b. (1 ) 1 3 1 4 1 5 1 6k

k

8

0

1 1 1 1 1 1 1 1 1 1c. 2.71828

! 1 1 2 6 24 120 720 5040 40,320n n

Properties of Sums

1

1. , is a constantn

i

c cn c

1 1

2. , is a constantn n

i ii i

ca c a c

1 1 1

3.n n n

i i i ii i i

a b a b

1 1 1

4.n n n

i i i ii i i

a b a b

SeriesMany applications involve the sum of the terms of a finite or an infinite sequence. Such a sum is called a series.

1

Example 9.

3For the series , find a the third partial sum and b the sum.

10i

3

1 2 31

3 3 3 3a. .3 .03 .003 0.333

10 10 10 10ii

1 2 3 4 51

3 3 3 3 3 3 1b. .3 .03 .003 .0003 .00003 0.33333

10 10 10 10 10 10 3ii

Notice that the sum of an infinite series can be a finite number.

Variations in the upper and lower limits of summation can produce quite different-looking summation notation for the same sum.

Sequences have many applications in situations that involve a recognizable pattern.

2

Example 10.

From 1970 to 2001, the resident population of the United States can be approximated by the

model 205.7 1.78 0.025 , 0,1,...,31 where is the population in millions

and represents thn na n n n a

n

e year, with 0 corresponding to 1970.

Find the last five terms of this finite sequence.

n

2

27 205.7 1.78 27 0.025 27 272.0a

2

28 205.7 1.78 28 0.025 28 275.1a

2

29 205.7 1.78 29 0.025 29 278.3a

2

30 205.7 1.78 30 0.025 30 281.6a 2

31 205.7 1.78 31 0.025 31 284.9a