A sequence is a function whose domain is the set of positive integers.

Definition of Sequence:

An infinite sequence is a function whose domain is the set of positive integers. The function values

a

_{1}, a

_{2}, a

_{3}, a

_{4}, ... , a

_{n}, ...

are the terms of the sequence. If the domain of the function consists of the first n positive integers only, the sequence is a finite sequence.

A. Finding the Terms of a Sequence:

Example 1: find the first five terms of a

_{n}= 4n - 7

a

_{1}= 4(1) - 7 = -3

a

_{2}= 4(2) - 7 = 8 - 7 = 1

a

_{3},= 4(3) - 7 = 12 - 7 = 5

a

_{4}= 4(4) - 7 = 16 - 7 = 9

a

_{5}= 4(5) - 7 = 20 - 7 = 13

Example 2: Find the 16th term of the sequence a

_{n}= (-1)

^{n-1}(n(n-1))

a

_{16}= (-1)

^{16-1}(16(16-1)) = (-1)

^{15}(16(15)) = (-1)(240) = -240

Example 3: Write the first five terms of the sequence defined recursively:

a

_{1}= 15, a

_{k+1}= a

_{k}+ 3

Let k = 1 so we have:

a

_{1+1}= a

_{2}= a

_{1}+ 3 = 15 + 3 = 18

Let k = 2 so we have:

a

_{2+1}= a

_{3}= a

_{2}+ 3 = 18 + 3 = 21

a

_{3+1}= a

_{4}= a

_{3}+ 3 = 21 + 3 = 24

a

_{4+1}= a

_{5}= a

_{4}+ 3 = 24 + 3 = 27

B. Finding the nth term of a Sequence

Example 4: Write an expression for the apparent nth term of the sequence

(assume n begins with 1): 3, 7, 11, 15, 19, ...

7 - 3 = 11 - 7 = 15 - 11 = 4

As you can see, the terms are going up by 4 and the first term is one less than 4 so

a

_{n}= 4n - 1

Example 5: 1, 1/4, 1/9, 1/16, 1/25, ...

As you can see, the terms denominators are perfect squares so

a

_{n}= 1/(n

^{2}) = n

^{-2}

C. The Fibonacci Sequence: A Recursive Sequence

The Fibonacci sequence is defined recursively as follows.

a

_{0}= 1, a

_{1}= 1, a

_{k}= a

_{k-2}+ a

_{k-1}, where k is greater than or equal to 2

{1, 1, 2, 3, 5, 8, ...}

Example 6: Write the first five terms of the sequence defined recursively. Use this pattern to write the nth term of the sequence as a function of n.

a

_{1}= 25, a

_{k+1}= a

_{k}- 5

a

_{2}= a

_{1+1}= a

_{1}-5 = 25 -5 = 20

a

_{3}= a

_{2+1}= a

_{2}-5 = 20 - 5 = 15

a

_{4}= a

_{3+1}= a

_{3}-5 = 15 - 5 = 10

a

_{5}= a

_{4+1}= a

_{4}-5 = 10 - 5 = 5

Therefore a

_{n}= 25 - 5n

C. Definition of Factorial:

In n is a positive integer, n factorial is defined by

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

As a special case, zero factorial is defined as 0! = 1

Example 7: Simplify the ratio of factorials.

(4!)/(7!) = (4!) / (7 x 6 x 5 x 4!) = 1/ (7 x 6 x 5 ) = 1 / 210

Example 8: Simplify the ratio of factorials.

(n + 2)! / (n!) = (n + 2)(n + 1)(n!) / (n!) = (n + 2)(n + 1) = n

^{2}+ 3n + 2

D. Definition of Summation Notation:

The sum of the first n terms of a sequence is represented by

The summation of a

_{i}when i = 1 to i = n is equal to

a

_{1}+ a

_{2}+ a

_{3}+ a

_{4}+ ... + a

_{n}

where

*i*is called the index of summation,

*n*is the upper limit of summation, and 1 is the lower limit of summation. (check out http://www.cs.fsu.edu/~cop4531/slideshow/chapter3/3-1.html)

Example 9: Find the sum of 3i - 1 when i = 1 to i = 6

3 - 1 + 6 - 1 + 9 - 1 + 12 - 1 + 15 - 1 + 18 - 1 = 57

To use your TI - 83, TI - 83 plus calculators:

Go to mode - change from function to sequential mode, then

2nd Stat arrow to math #5 Sum, 2nd Stat arrow to ops #5 seq then put in the sequence, then n to show the calculator the variable, then the lower bound number, then the upper bound number, then close the parenthesis twice, then press enter.

Your screen should look like this:

sum(seq(3n-1,n,1,6)) = 57

E. Properties of Sums - http://www.libraryofmath.com/summation-formulas.html

F. Series:

Definition of a Series:

Consider the infinite sequence a

_{1}, a

_{2}, a

_{3}, a

_{4}, ... , a

_{i}, ...

1. The sum of all terms of the infinite sequence is called an

**infinite series**and is denoted by

a

_{1}+ a

_{2}+ a

_{3}+ a

_{4}+ ... + a

_{i}+ ... = The summation of a

_{i}when i = 1 to i = infinity.

2. The sum of the first n terms of the sequence is called a finite series or the nth partial sum of the sequence and is denoted by

a

_{1}+ a

_{2}+ a

_{3}+ a

_{4}+ ... + a

_{n}= The summation of a

_{i}when i = 1 to i = n.

Example 10: Find the sum of the partial sum of the series:

The summation of 8(-1/2)

^{n}when n = 1 to n = infinity, the 4th partial sum

= 8(-½ )

^{1}+ 8(-½ )

^{2}+ 8(-½ )

^{3}+ 8(-½ )

^{4}= -4 + 2 + -1 + .5 = -5/2

(Calculator check: you can check your individual answers by putting you calculator in function mode, put the series into y1 = 8 (-.5)

^{x}and looking in your table to get your values.)

Example 11: Find the sum of the infinite series:

The summation of 4(1/10)

^{n}when n = 1 to n = infinity

= 4 [ .1 + .01 + .001 + .0001 + ...] = 4 (.111111...) = 4 (1/9) = 4/9

Example 12: Find the sum of the infinite series:

The summation of 8(1/10)

^{n}when n = 1 to n = infinity is equal to

= 8 [ .1 + .01 + .001 + .0001 + ...] = 8 (.111111...) = 8 (1/9) = 8/9