An infinite geometric progression has sum 20, and the sum of the squares of its terms is 100. What is its first term?

Let the first term be a and the common ratio r (|r| < 1).

a/(1 − r) = 20 …(i)
a²/(1 − r²) = 100 …(ii)

Dividing (ii) by (i): a/(1 + r) = 5 …(iii)

From (i) and (iii): 20(1 − r) = 5(1 + r) → 20 − 20r = 5 + 5r → r = 3/5.
Then a = 20 × 2/5 = 8.

Check: squares form a GP with first term 64 and ratio 9/25, so the sum = 64/(16/25) = 100. ✓