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Question 210033
JEE Mathematics · Hard

Sequences and Series: AGP sum MCQ

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AGP sumHardQuestion 210033

Question

The sum 1 + 2·3 + 3·3² + 4·3³ + ... + n·3^(n−1) equals:
  1. A
    (2n−1)3ⁿ + 1)/4
  2. B
    ((2n−1)3ⁿ+1)/4
    Correct
  3. C
    3ⁿ(n−1)+1
  4. D
    n·3ⁿ/2

Correct answer

((2n−1)3ⁿ+1)/4

Explanation

This is an arithmetico-geometric series. Use S − 3S method: S = Σ r·3^(r−1). Then S − 3S = 1+3+9+...+3^(n−1) − n·3ⁿ = (3ⁿ−1)/2 − n·3ⁿ. Solve: S = ((2n−1)3ⁿ+1)/4.

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