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 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

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 $r.$

Penny

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