Numerical answer
An infinite geometric progression with positive first term has sum 20, and the sum of the squares of its terms is 100. What is its first term?
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Answer: 8
a/(1 − r) = 20 … (1) and a²/(1 − r²) = 100 … (2).
Divide (2) by the square of (1):
[a²/((1 − r)(1 + r))] ÷ [a²/(1 − r)²] = 100/400, so (1 − r)/(1 + r) = 1/4.
This gives 4 − 4r = 1 + r, so r = 3/5.
a = 20(1 − 3/5) = 8.
Check: squares form a GP with first term 64 and ratio 9/25, so the sum is 64/(16/25) = 100. ✓
Divide (2) by the square of (1):
[a²/((1 − r)(1 + r))] ÷ [a²/(1 − r)²] = 100/400, so (1 − r)/(1 + r) = 1/4.
This gives 4 − 4r = 1 + r, so r = 3/5.
a = 20(1 − 3/5) = 8.
Check: squares form a GP with first term 64 and ratio 9/25, so the sum is 64/(16/25) = 100. ✓
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