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Let Sₙ be the sum of the first n terms of a geometric progression with common ratio r. Then the value of (r-1) (dSₙ)/(dr) is

1. (n-1)Sₙ+nSₙ₋₁
2. (n-1)Sₙ-nSₙ₋₁
3. (n-1)Sₙ
4. None of these

Correct answer: (n-1)Sₙ-nSₙ₋₁

Solution

The expression for the derivative of the sum of a geometric series with respect to the common ratio captures how changes in the ratio affect the sum. The correct option reflects the relationship between the sums of the series at different terms, specifically accounting for the contributions from both the current and previous sums.

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