Single correct option
The sum of an infinite geometric progression with positive terms is 20, and the sum of the squares of its terms is 100. What is the first term of the progression?
- A. 6
- B. 5
- C. 10
- D. 8
Show answer and solution
Answer: (D) 8
Let the first term be a and common ratio r (0 < r < 1).
a/(1 − r) = 20 … (i)
a²/(1 − r²) = 100 … (ii)
Divide (ii) by (i): a/(1 + r) = 5, so a = 5(1 + r).
From (i): a = 20(1 − r).
5 + 5r = 20 − 20r → 25r = 15 → r = 3/5, and a = 20 × 2/5 = 8.
Check: 8/(2/5) = 20 ✓; 64/(1 − 9/25) = 64 × 25/16 = 100 ✓.
a/(1 − r) = 20 … (i)
a²/(1 − r²) = 100 … (ii)
Divide (ii) by (i): a/(1 + r) = 5, so a = 5(1 + r).
From (i): a = 20(1 − r).
5 + 5r = 20 − 20r → 25r = 15 → r = 3/5, and a = 20 × 2/5 = 8.
Check: 8/(2/5) = 20 ✓; 64/(1 − 9/25) = 64 × 25/16 = 100 ✓.
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