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NDA 2010
Exam
NDA
Year
2010
Subject
Mathematics
Chapter
Sequences and Series

NDA 2010 · Mathematics · Sequences and Series

If p times the p th term of an AP is q times the q th term, then...

If pp times the pp th term of an AP is qq times the qq th term, then what is the (p+q)(p+q) th term equal to?

Options

  1. Ap+qp+q
  2. Bpqp q
  3. C1
  4. D0

Correct answer

D. 0

Step-by-step solution

pth=a+(p−1)dqth=a+(q−1)d\begin{aligned} p^{t h} & =a+(p-1) d \\ q^{t h} & =a+(q-1) d \end{aligned} By hypothesis ap+(p2−p)d=aq+(q2−q)d(ap−aq)=(q2−q−p2+p)d or  a(p−q)=[(p−q)−(p2−q2)]d(p+q) th =a+(p+q−1)d=[1−(p+q)]d−[1−(p+q)]d=0\begin{aligned} & a p+\left(p^2-p\right) d=a q+\left(q^2-q\right) d \\ & (a p-a q)=\left(q^2-q-p^2+p\right) d \\ & \text{\text{ or } } a(p-q)=\left[(p-q)-\left(p^2-q^2\right)\right] d \\ & (p+q) \text{ th }=a+(p+q-1) d \\ & =[1-(p+q)] d-[1-(p+q)] d=0 \end{aligned}
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