Geometric Series Formula

Before knowing the geometric series formula, let us recall what is a geometric series. It is a series (sum of terms) where the ratios of every two consecutive terms are the same. The geometric series formula includes:

• Formula to find the n^th term of a geometric series.
• Finding the sum of a finite geometric series.
• Finding the sum of an infinite geometric series.

What is a Geometric Series?

A geometric series is the sum of finite or infinite terms of a geometric sequence. For the geometric sequence a, ar, ar², ..., ar^n-1, ..., the corresponding geometric series is a + ar + ar² + ..., ar^n-1 + .... We know that "series" means "sum". In particular, the geometric series means the sum of the terms that have a common ratio between every adjacent two of them. There can be two types of geometric series: finite and infinite. Here are some examples of geometric series.

• 1/2 + 1/4 + .... + 1/8192 is a finite geometric series
  where the first tern, a = 1/2 and the common ratio, r = 1/2
• -4 + 2 - 1 + 1/2 - 1/4 + ... is an infinite geometric series
  where the first term, a = -4 and the common ratio r = -1/2

Geometric Series Formula

The geometric series formula refers to the formula that gives the sum of a finite geometric sequence, the sum of an infinite geometric series, and the n^th term of a geometric sequence. The sequence is of the form {a, ar, ar², ar³, …….} where, a is the first term, and r is the "common ratio".

Geometric Series Formulas

The formulas for a geometric series include the formulas to find the n^th term, the sum of n terms, and the sum of infinite terms. Let us consider a geometric series whose first term is a and common ratio is r.

a + ar + ar² + ar³ + ...

Formula 1: The n^th term of a geometric sequence is,

n^th term = a r^n-1

Where,

• a is the first term
• r is the common ratio of every two successive terms
• n is the number of terms.

To see how this formula is derived, click here.

Formula 2: The sum formula of a finite geometric series a + ar + ar² + ar³ + ... + a r^n-1 is

Sum of n terms = a (1 - rⁿ) / (1 - r) (or) a (rⁿ - 1) / (r - 1)

where,

• a is the first term
• r is the common ratio every two consecutive terms
• n is the number of terms.

To see how this formula is derived, click here.

Formula 3: The sum formula of an infinite geometric series a + ar + ar² + ar³ + ... is

Sum of infinite geometric series = a / (1 - r)

where,

• a is the first term
• r is the common ratio every two successive terms

To see how this formula is derived, click here.

Convergence of Geometric Series

A finite geometric series always converges. But the convergence of an infinite geometric series depends upon the value of its common ratio. An infinite geometric series a, ar, ar², ...

• converges when |r| < 1 and hence we can find its sum using the formula a / (1 - r).
• diverges when |r| > 1 and hence we can't find its sum in this case.

Examples:

The geometric series 2 + 4 + 8 + 16 + .... diverges as |r| = |2| = 2 > 1. So its sum can't be found.

The geometric series 1 - 1/2 + 1/4 - 1/8 + 1/16 - .... converges as |r| = |-1/2| = 1/2 < 1. Hence, its sum is, a / (1 - r) = 1 / (1 - (-1/2)) = 1/ (3/2) = 2/3.