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NDA 2 2024 Maths, question 38

Topic: Algebra

Q38

In an AP, the ratio of the sum of the first pp terms to the sum of the first qq terms is p2:q2p^{2}:q^{2}. Which one of the following is correct?

  1. (a) The first term is equal to the common difference
  2. (b) The first term is equal to twice the common difference
  3. (c) The common difference is equal to twice the first termCorrect answer
  4. (d) The first term is equal to square of the common difference

Solution.

Answer: (c) The common difference is equal to twice the first term

  1. SpSq=p[2a+(p−1)d]q[2a+(q−1)d]=p2q2\frac{S_{p}}{S_{q}} = \frac{p[2a + (p - 1)d]}{q[2a + (q - 1)d]} = \frac{p^{2}}{q^{2}}, so q[2a+(p−1)d]=p[2a+(q−1)d]q[2a + (p - 1)d] = p[2a + (q - 1)d].
  2. This simplifies to (q−p)(2a−d)=0(q - p)(2a - d) = 0, and with p≠qp \neq q, d=2ad = 2a.

Trap: If p=qp = q the ratio says nothing; the question means two different values pp and qq.