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NDA 1 2025 Maths, question 1

Topic: Algebra

Q1

The sum of the first kk terms of a series SS is 3k2+5k3k^{2} + 5k. Which one of the following is correct?

  1. (a) The terms of S form an arithmetic progression with common difference 14.
  2. (b) The terms of S form an arithmetic progression with common difference 6.Correct answer
  3. (c) The terms of S form a geometric progression with common ratio 10/7.
  4. (d) The terms of S form a geometric progression with common ratio 11/4.

Solution.

Answer: (b) The terms of S form an arithmetic progression with common difference 6.

  1. The kkth term is the difference of consecutive partial sums: ak=Sk−Sk−1=3k2+5k−[3(k−1)2+5(k−1)]=6k+2a_{k} = S_{k} - S_{k - 1} = 3k^{2} + 5k - \left\lbrack 3(k - 1)^{2} + 5(k - 1) \right\rbrack = 6k + 2.
  2. Therefore ak+1−ak=6a_{k + 1} - a_{k} = 6, so the terms form an AP with common difference 6.