9.2 Arithmetic Sequences and Partial Sums

This section covers:

• How to find the nth terms of arithmetic sequences
• How to find the sums of arithmetic sequences

       Arithmetic Sequences

A sequence of terms is considered arithmetic if the differences between consecutive terms are the same.

The following sequence:
         a₁, a₂, a₃, a₄, ..., a_n, ...
is arithmetic if there is a number d such that a₂ - a₁ = a₃ - a₂ = a₄ - a₃ = ... = d.

The number d is the common difference of the arithmetic sequence.

An example of an arithmetic sequence is:
         3, 7, 11, 15, 19, ...
because:
         7 - 3 = 4
         11 - 7 = 4
         15 - 11 = 4
         19 - 15 = 4
In this case, d, the common difference, is 4.

The nth term of an arithmetic sequence has the form:
         a_n = a₁ + d(n - 1).

For example:

In the sequence -1, 12, 25, 38, 51, where -1 = a₂, find the 9th term in the sequence.
         a₉ = a₁ + d(n - 1)
         12 - (-1) = 13, so d = 13.
         a₂ - d = a₁
         -1 - 13 = -14 = a₁
         a₉ = (-14) + 13(9 - 1)
         a₉ =  90.In this sequence, the 9th term is 90.

If you know the nth term of an arithmetic sequence and you know the common difference of the sequence, you can find the (n+1)th term by using the recursive formula:
         a_n+1 = a_n + d
              or
         a_n = a₁ + (n-1)d


       Sums and Partial Sums

The sum of a finite arithmetic sequence with n terms is: S_n = ⁿ/₂(a₁ + a_n).
For example:

In a sequence of 7 terms where a₁ = 56, d = 23, and a₇ = 194, find the sum of the sequence.
S_n = ⁿ/₂(a₁ + a_n)
S₇ = ⁷/₂(56+194)
S₇ = 875

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