If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to

Solution

Maths sir · 3 steps

Every even-place term is just r times the odd-place term before it, so the total is (1 + r) times the odd sum.

1. Split the terms. Group the 64 terms into 32 terms in odd places, a, ar², ar⁴, …, and 32 terms in even places, ar, ar³, ar⁵, …. Call the sum of the odd-place terms S_(odd).
2. Pair them up. Each even-place term is r times the odd-place term just before it, so the even-place sum is r S_(odd). The total is then S_(odd) + r S_(odd) = (1 + r) S_(odd).

   (1 + r) S_(odd) = 7 S_(odd)

3. So. Since S_(odd) ≠ 0, divide both sides by it: 1 + r = 7, so r = 6. That is option (b).

The idea to keep. In a G.P. with an even number of terms, the even-place sum is r times the odd-place sum. You never need the sum formula, only this pairing.

Answer from the NTA final answer key. Solution by PrepWiser.