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NDA 2012
Exam
NDA
Year
2012
Subject
Mathematics
Chapter
Sequences and Series

NDA 2012 · Mathematics · Sequences and Series

If p, q, r are in AP as well as GP, then which one of the following...

If p,q,rp, q, r are in AP as well as GP, then which one of the following is correct?

Options

  1. Ap=q≠rp=q \neq r
  2. Bp≠q≠rp \neq q \neq r
  3. Cp≠q=rp \neq q=r
  4. Dp=q=rp=q=r

Correct answer

D. p=q=rp=q=r

Step-by-step solution

 A.P. →2q=p+r....(1) G.P. →q2=pr⇒0=pr2pr=p+rp+r−2pr=0(p)2+(r)2−2pr=0(p−r)2=0p=rp=r....(2)2q=2pq=p....(3)\begin{aligned} & \text{ A.P. } \rightarrow 2 q=p+r....(1) \\ & \text{ G.P. } \rightarrow q^2=p r \Rightarrow 0=\sqrt{p r} \\ & 2 \sqrt{p r}=p+r \\ & p+r-2 \sqrt{p r}=0 \\ &(\sqrt{p})^2+(\sqrt{r})^2-2 \sqrt{p} \sqrt{r}=0 \\ &(\sqrt{p}-\sqrt{r})^2=0 \\ & \sqrt{p}=\sqrt{r} \\ & p=r....(2) \\ & 2 q=2 p \\ & q=p....(3) \end{aligned} from (2) and (3) p=q=r.p=q=r .
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