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Year 11 · Patterns & Algebra

Geometric sequences and series

An arithmetic sequence adds the same amount each time; a geometric sequence multiplies by the same amount.

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How to do it

  1. Arithmetic: nth term = a + (n − 1)d, sum = n/2 (2a + (n − 1)d).
  2. Geometric: nth term = a × rⁿ⁻¹, sum = a(rⁿ − 1) ÷ (r − 1).

Worked example

Find the sum of the first 5 terms of 4, -8, 16, -32, …
Geometric with a = 4, r = -2. Sₙ = a(rⁿ − 1)/(r − 1) = 4(-2^5 − 1)/(-2 − 1) = 44.
Answer: 44

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Use n − 1, not n, in the nth-term formula.

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