Series and SequencesAn infinite sequence is an unending list of numbers that follow a pattern. The terms of the sequence are written a

1, a

2, a

3,...,a

n,...

If the list ends, we call it a finite sequence.

Ex. Write the first four terms of the sequence:

a) an = 3n – 2

b) an = 3 + (-1)n

Ex. Write the first four terms of the sequence 1

2 1

n

nan

Ex. Write an expression for an:

a) 1, 3, 5, 7,...

b) 2, -5, 10, -17,...

A sequence is recursive if each term is defined by one or more previous terms

Ex. The Fibonacci sequence is defined as a0 = 1, a

1 = 1, a

k = a

k – 1 + a

k

– 2. Write the first six terms.

Ex. Find the first five terms of the recursive sequence defined by a1 = 3, ak = 2ak – 1 – 5

If n is a positive integer, n factorial is defined as

n! = 1 ∙ 2 ∙ 3 ∙ 4 ∙ ... ∙ n

As a special case, 0! = 1.

Keep in mind that parentheses matter:

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

(2n)! = 1 ∙ 2 ∙ 3 ∙ 4 ∙ ... ∙ 2n

Ex. Write the first five terms of the sequence2

!

n

nan

Ex. Evaluate the factorial

a)

b)

c)

8!

2! 6!

2! 6!

3! 5!

1 !

1 !

n

n

The Greek letter sigma (Σ) can be used to show the sum of many terms

i is called the index of the summation

n is the upper limit of the summation

1 is the lower limit of the summation

1 2 31

...n

i ni

a a a a a

Ex. Find the sum

a)

b)

c)

5

1

3i

i

6

2

3

1k

k

8

0

1

!i i

Consider the infinite sequence a1, a2, a3,..., ai,...

The sum of the first n terms is called the nth partial sum of the sequence, and is denoted

The sum of all the terms of the infinite sequence is called an infinite series, and is denoted

1 2 31

...n

i n ni

a a a a a S

1 2 31

... ...i ii

a a a a a

Ex. Use the first 3 partial sums to evaluate the sum1

3

10ii

Practice Problems

Section 8.1

Problems 1, 17, 37, 51, 59, 67, 73, 99

Arithmetic Sequences and SeriesA sequence is arithmetic if the difference of two consecutive terms is the same.

an + 1 – an = d for any positive integer n

The number d is called the common difference

Ex. Find the first 4 terms of the arithmetic sequence.

a) an = 4n + 3

b) an = 7 – 5n

c) 14 3na n

To find the nth term of an arithmetic sequence, we use the formula

an = a

1 + d(n – 1)

where a1 is the first term and d is the

common difference

Ex. Find the nth term of the sequence2, 5, 8, 11, 14,...

Ex. The fourth term of an arithmetic sequence is 20 and the 13th term is 65. Find the nth term.

Ex. Find the 9th term of the arithmetic sequence that starts with 2 and 9.

To find the sum of a finite arithmetic sequence with n terms, we use the formula

Ex. Find the sum of the first 10 odd numbers.

Ex. Find the sum of the integers from 1 to 100

12n

n nS a a

Ex. Find the 150th partial sum of the arithmetic sequence5, 16, 27, 38, 49,...

Ex. Find the sum100

51

7n

n

Ex. In a golf tournament, 16 golfers win cash prizes. First place gets $1000, second place gets $950, third place gets $900, and so on. What is the total amount of prize money?

Practice Problems

Section 8.2

Problems 1, 21, 37, 45, 63, 65, 69, 89

Geometric Sequences and SeriesA sequence is geometric if the quotient of two consecutive terms is the same.

for any positive integer n

The number r is called the common ratio

1n

n

ar

a

Ex. Find the first 4 terms of the geometric sequence.

a) an = 2n

b) an = 4(3n)

c)

d) an = n2

13

n

na

To find the nth term of a geometric sequence, we use the formula

an = a

1rn – 1

where a1 is the first term and r is the

common ratio

Ex. Find the nth term of the sequence3, 6, 12, 24, 48,...

Ex. Write the 15th term of the geometric sequence whose 1st term is 20 and whose common ratio is 1.05.

Ex. Write the 12th term of the geometric sequence 5, 15, 45,...

Ex. The 4th term of a geometric sequence is 125 and the 10th term is . Find the 14th term.

12564

To find the sum of finite geometric sequence with n terms, we use the formula

Ex. Find the sum

1

1

1

n

n

rS a

r

12

1

2

4 1.03i

i

It is possible to take the sum of an infinite geometric sequence and get a finite answer.

Consider

We say that this geometric series converges

A geometric series will converge if |r| < 1, and the sum is given by the formula

1

1

aS

r

1 1 1 12 4 8 16

Ex. Find the sum

a)

b) 0.3 + 0.03 + 0.003 +...

1

1

4 .06n

n

Ex. A deposit of $50 is made on the first day of every month in an account that pays 6% compounded monthly. What is the balance at the end of 2 years?

Practice Problems

Section 8.3

Problems 1, 11, 27, 35, 41, 57, 79, 107