Geometric sequences and series

Objective. Model repeated multiplicative change and finite sums.

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2. Worked example
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Learn: the key idea

A geometric sequence multiplies each term by a constant common ratio r to get the next, with nth term aₙ=ar^(n−1), and the sum of the first n terms is Sₙ=a(1−rⁿ)/(1−r) for r≠1. When |r|<1, the sequence converges and has an infinite sum S∞=a/(1−r), modelling situations like repeated halving. For a=100 and r=0.5, the 4th term is 100×0.5³=12.5, and the infinite sum is 100/0.5=200. Always check whether r is between −1 and 1 before using the infinite sum formula.

Worked example

Find the sum to infinity of the geometric series with first term 12 and common ratio 1/3.

1 $Check |r|=1/3<1, so the infinite sum exists$
2 $Use S\infty = a/(1-r)$
3 $Substitute: S\infty = 12/(1-1/3) = 12/(2/3)$
4 $Simplify to get S\infty = 18$

Answer: 18

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Common mistake. Students apply the infinite sum formula to a sequence with r=2, which diverges; the infinite sum formula only applies when |r|<1, otherwise the sum grows without bound and has no finite value.

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