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Algebra · SS 2

Sum to infinity of a GP

Find what an endless GP adds up to when its terms keep shrinking.

  • S∞ = a1 − r (only when −1 < r < 1)

What the letters mean

a
the first term
r
the common ratio, between −1 and 1

Worked example

Find 8 + 4 + 2 + 1 + … forever.

  1. Step 1: a = 8, r = 12

    S∞ = 81 − 0.5 = 80.5

  2. Step 2: Divide

    = 16

Final answer: 16

See it

Play with this picture from the Geometric Progression lesson.

2T16T218T354T4162T5486T61,458T74,374T8

Each bar = the bar before × 3

T4 = a × r3 = 2 × 33 = 2 × 27 = 54

Sum of the first 4 terms: S4 = 80

Term number n4
First term a2
Common ratio r3