Geometric Sequences

A geometric sequence (or geometric progression, GP) is an ordered set of numbers in which the ratio between consecutive values is constant. For example:

2, 6, 18, 54, 162, 486, …

The sum of a sequence is called a series. For example:

2 + 6 + 18 + 54 + 162 + 486 + …

The general term in a geometric sequence is given by

a_k = ar^{k-1}

The sum of the first n terms of a geometric sequence is given by

S_n = sum{k=1}{n}{ar^{k-1}} = a{1-r^n}/{1-r}

In a convergent series, the sum of the series (an infinite number of terms) is given by

S = a/{1-r}

where

  • a = first term
  • ak = kth term
  • r = common ratio
  • k = term number
  • n = number of terms
  • Sn = sum of first n terms of sequence