Arithmetic and geometric series

A series adds up the terms of a sequence. Arithmetic series add the same amount each time; geometric series multiply by the same ratio. A geometric series with −1 < r < 1 even has a finite sum when it goes on for ever.

Key facts

  • Arithmetic: uₙ = a + (n − 1)d, Sₙ = n/2 (2a + (n − 1)d).
  • Geometric: uₙ = arⁿ⁻¹, Sₙ = a(1 − rⁿ)/(1 − r).
  • Sum to infinity: S∞ = a/(1 − r) when |r| < 1.
  • Σ (sigma) means “add up”.

Worked examples

Example 1

Find the sum of the first 20 terms of 5 + 8 + 11 + ….

  1. a = 5, d = 3, n = 20.
  2. S = 20/2 × (10 + 19 × 3) = 10 × 67 = 670

Answer: 670

Example 2

Find the sum to infinity of 12 + 6 + 3 + ….

  1. a = 12, r = ½.
  2. S∞ = 12/(1 − ½) = 24

Answer: 24