find the common ratio

Question

find the common ratio of an infinite geometric series with the given sum and first term s=18 a1=9

Answer 1

Hi Erika C.,

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Associated with every geometric sequence {𝑎𝑟^𝑛−1} is an infinite geometric series

Σ 𝑎𝑟^𝑛−1 = 𝑎 + 𝑎𝑟 + 𝑎𝑟² + . . . + 𝑎𝑟^𝑛−1+ . . .

If -1 < r < 1, then the series converges to 𝑎₁/(1 − 𝑟) , and we write

Σ = 𝑎𝑟^𝑛−1 = 𝑎 + 𝑎𝑟 + 𝑎𝑟² + . . . + 𝑎𝑟^𝑛−1 + . . . = 𝑎₁/(1 − 𝑟)

If |r| ≥ 1, then the infinite series does not have a sum, and the series diverges.

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The question states this is a infinite geometric series, and it converges to a sum (18) that is equal to 9/(1 - r).

We therefore have 18 = 9/(1 - r), I'll leave it to you to solve for r (the common ratio).

I hope this helps, Joe.