Geometric Series

A geometric series is the sum of the terms in a geometric sequence. If the sequence has a definite number of terms, the simple formula for the sum is

Formula 3: equation

This form of the formula is used when the number of terms ( n), the first term ( a 1), and the common ratio ( r) are known.

Another formula for the sum of a geometric sequence is

Formula 4: equation

This form requires the first term ( a 1), the last term ( a n ), and the common ratio ( r) but does not require the number of terms ( n).

Example 1

Find the sum of the first five terms of the geometric sequence in which a 1 = 3 and r = –2.

a 1 = 3, r = –2, n = 5

Use Formula 4: equation

Example 2

Find the sum of the geometric sequence for which equation.

equation

Use Formula 4: equation

Example 3

Find a 1 in each geometric series described.

  1. S n = 244, r = –3, n = 5

  2. S n = 15.75, r = 0.5, a n = 0.25

  1. S n = 244, r = –3, n = 5

    Use Formula 3: equation

  2. S n = 15.75, r = 0.5, a n = 0.25

    Use Formula 4: equation

Formula 5:

If a geometric series is infinite (that is, endless) and –1 < r < 1, then the formula for its sum becomes equation

If r > 1 or if r < –1, then the infinite series does not have a sum.

Example 4

Find the sum of each of the following geometric series.

  1. 25 + 20 + 16 + 12.8 + …

  2. 3 – 9 + 27 – 81 + …

  1. 25 + 20 + 16 + 12.8 + …

    First find r.

    equation

    Since equation, this infinite geometric series has a sum.

    Use Formula 5.

    equation

  2. 3 – 9 + 27 – 81 + …

    First find r.

    equation

    Since –3 < –1, this geometric series does not have a sum.