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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Sequences and Series

NDA 2024 · Mathematics · Sequences and Series

In an A P , the ratio of the sum of the first p terms to the sum of...

In an APA P, 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?

Options

  1. AThe first term is equal to the common difference
  2. BThe first term is equal to twice the common difference
  3. CThe common difference is equal to twice the first term
  4. DThe first term is equal to square of the common difference

Correct answer

C. The common difference is equal to twice the first term

Step-by-step solution

Wehave,We have,SpSq=p2q2⇒p2[2a+(p−1)d]q2[2a+(q−1)d]=p2q2{∵Sn=n2[2a+(n−1)d]}⇒2a+(p−1)d2a+(q−1)d=pq⇒2aq+q(p−1)d=2ap+p(q−1)d⇒2a(q−p)=d(q−p)⇒2a=d or  d=2a\begin{aligned} & \frac{S_p}{S_q}=\frac{p^2}{q^2} \\ & \Rightarrow \frac{\frac{p}{2}[2 a+(p-1) d]}{\frac{q}{2}[2 a+(q-1) d]}=\frac{p^2}{q^2}\left\{\because S_n=\frac{n}{2}[2 a+(n-1) d]\right\} \\ & \Rightarrow \frac{2 a+(p-1) d}{2 a+(q-1) d}=\frac{p}{q} \\ & \Rightarrow 2 a q+q(p-1) d=2 a p+p(q-1) d \\ & \Rightarrow 2 a(q-p)=d(q-p) \\ & \Rightarrow 2 a=d \text{\text{ or } } d=2 a \end{aligned}$$ This, the common difference is equal to twice of the first term.
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