Geometric series

A geometric series is the sum of a sequence that is multiplied by the same number, the common ratio r, each time. When r is between −1 and 1, the terms shrink so fast that the series has a sum to infinity.

Key facts

  • Geometric: uₙ = arⁿ⁻¹ and Sₙ = a(1 − rⁿ)/(1 − r).
  • If −1 < r < 1, a geometric series converges, with sum to infinity S∞ = a/(1 − r).
  • a is the first term and r the common ratio: divide any term by the one before it.

Worked examples

Example 1

Find the sum of the first 8 terms of 3 + 6 + 12 + …

  1. Use a = 3, r = 2 and n = 8. S = 3(2⁸ − 1)/(2 − 1)
  2. Work it out. 3 × 255 = 765

Answer: 765

Example 2

Find the common ratio of the geometric sequence 81, 27, 9, …

  1. Divide the second term by the first. 27 ÷ 81 = 1/3
  2. Check with the next pair. 9 ÷ 27 = 1/3

Answer: 1/3

Example 3

Find the 6th term of the geometric sequence 2, 6, 18, …

  1. Use a = 2 and r = 3. u₆ = 2 × 3⁵
  2. Work it out. 2 × 243 = 486

Answer: 486

Example 4

Evaluate Σ 3(2ʳ) from r = 1 to 10.

  1. The first term is 3 × 2 = 6 and r = 2. S = 6(2¹⁰ − 1)/(2 − 1)
  2. Work it out. 6 × 1,023 = 6,138

Answer: 6,138

Example 5

Find the sum to infinity of 50 + 25 + 12.5 + …

  1. Use a = 50 and r = 0.5; −1 < r < 1, so the series converges. S∞ = 50/(1 − 0.5)
  2. Work it out. 50 ÷ 0.5 = 100

Answer: 100

Example 6

A geometric series has first term 24 and sum to infinity 36. Find the common ratio.

  1. Substitute. 36 = 24/(1 − r)
  2. Rearrange. 1 − r = 24/36 = 2/3
  3. Solve for r. r = 1/3

Answer: r = 1/3