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Year 11 · Patterns & Algebra

Sum to infinity

A geometric series with |r| < 1 shrinks towards a limit: its sum to infinity is a ÷ (1 − r).

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How to do it

  1. Find a (first term) and r (ratio).
  2. Check −1 < r < 1.
  3. S∞ = a ÷ (1 − r).

Worked example

Find the sum to infinity of 32 + 32/5 + 32/25 + …
a = 32, r = 1/5. S∞ = a ÷ (1 − r) = 32 ÷ 4/5 = 40.
Answer: 40

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If |r| ≥ 1 the sum does not exist.

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