geometric sequence

A sequence of numbers in which the ratio of consecutive terms is constant. For instance, the sequence 1, 2, 4, 8, 16, 32, ... is a geometric sequence.

A geometric series is one where each term is generated by applying a multiplier to the last. This is a geometric series with common difference *2 -

2 + 4 + 8 + 16 + 32

If we want to find the sum of this series (the sum is the total of all the terms added together), we use the following formula -

                            rn - 1
                a *   -----------------
                            r - 1

Where a is the first term, r is the common ratio, and n is the number of terms.

Example 1

Sum the series 2 + 6 + 18 + 54 + ... (8 terms).

First term = 2
Common difference = 3
Number of terms = 8

2 * ((3⁸ - 1)/(3 - 1)) = 6560.

Example 2

Sum the series 8 + 4 + 2 + 1 + 0.5 + ... (10 terms)

First term = 8
Common difference = 0.5
Number of terms = 10

8 * ((0.5¹⁰ - 1)/(0.5 - 1)) = 15.984

Note how geometric series which involve division can be modelled by substituting the appropriate fraction - dividing by 2 is the same as multiplying by 0.5.