# Sequences and Sums

A sequence is a function t from a subset of the integers
(usually N or Z+) to a set S. In this

discussion, the set S will be a set of numbers but we could have a sequence of
colors, musical notes,

or even computer programs.

We often write . Each
is a term of the
sequence. The sequence may

be denoted by or
. The numbers, 1, 2, 3,... are the term
indices and

are the term values

**1 Sequence Examples**

Can you give the next term of each of these sequences? Can you give the function
t(n) that defines

the sequence? Assume the first term is .

2 Some Special Sequences

2.1 Arithmetic Sequences

An arithmetic sequence is a sequence in which each term equals the preceding
term plus a constant.

A general arithmetic sequence looks like this a, a + d, a + 2d, a + 3d,... The
first term is a and the

constant difference between terms is d.

What is the n^{th} term of the sequence a, a + d, a + 2d, a + 3d, : : :?

What is the next term and the nth term of each of these sequences?

4, 7, 10, 13, 16, 19, ...

-7,-1, 5, 11, 17, 23, ...

**2.2 Geometric Sequences**

A geometric sequence is a sequence in which each term equals the preceding term
times a constant.

A general geometric sequence looks like this a, ar, ar2, ar3, ar4,... the first
term is a and the

constant ratio between successive terms is r.

What is the n^{th} term of the sequence a, ar, ar^{2}, ar^{3}, ar^{4},...?

What is the next term and the nth term of each of these sequences?

2, 6, 18, 54, 162, ...

1,-4, 16,-64, 256,...

**2.3 Quadratic Sequences**

A quadratic sequence is a sequence whose n^{th} term is given by a quadratic
function, .

Here are some quadratic sequences. Can you find the quadratic function that
generates them?

1, 4, 9, 16, 25, 36, ...

6, 15, 28, 45, 66, 91,...

An arithmetic sequence is given by a linear function and the difference between
successive terms is

a constant. In a quadratic sequence the differences between successive terms are
given by a linear

function and the second differences are constant. For example:

sequence | |

differences | |

second differences |

**3 Series and Partial Sums**

A series is a sum of the terms of a sequence. Since a sequence has infinitely
many terms, a series

is the sum of infinitely many terms. We often sum only the first n terms of a
sequence. The sum

of the first n terms is called the nth partial sum. The following is a sum of an
infinite number of

terms.

Below is the sum of the first n terms. This is the n^{th}
partial sum. k is the index of summation, 1

is the lower limit, n is the upper limit.

This is also a partial sum. j is the index of summation, m is the lower limit, n is the upper limit.

**3.1 Arithmetic Sums**

An arithmetic sum is a sum of terms of an arithmetic sequence

**3.1.1 Examples**

**3.2 Geometric Sums and Series**

A geometric sum is a sum of terms of a geometric sequence.

or, summing from 0

With geometric sequences, summing the infinite sequence
may be meaningful. infinite sums are

called series.

Let's find a formula fort the nth partial sum

Let

Then

as all the other terms
cancel out when we subtract rS from S.

**3.2.1 Examples**

.What happens when n gets very large?

**3.3 Some Special Sums**