Math CentralQuandaries & Queries


Question from Muhammad, a student:

The sum of an infinite geometric series is 15 and the sum of their squares is 45. Find the series

Hi Muhammad,

Suppose the infinite geometric series is

\[a + ar + ar^2 + ar^3 \cdot \cdot \cdot\]

If the series converges then its sum is $\large \frac{a}{1 - r}.$ This sum you know is 15. Now square each term to get the infinite geometric series

\[a^2 + a^2 r^2 + a^2 r^4 \cdot \cdot \cdot\]

What is the sum of this series? You know the value is 45. This gives you two equations in $a$ and $r.$ Solve for $a$ and $r.$


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