Q34 / 90 · JEE Main 2012 (26 May Online) · Mathematics

If the A.M. between $p^{\text {th }}$ and $q^{\text {th }}$ terms of an A.P. is equal to the A.M. between $r^{\text {th }}$ and $s^{\text {th }}$ terms of the same A.P., then $p+q$ is equal to

  • A
    $r+s-1$
  • B
    $r+s-2$
  • C
    $r+s+1$
  • Correct answer
    D
    $r+s$

Solution

Given: $\frac{a_p+a_q}{2}=\frac{a_r+a_s}{2}$ $$ \begin{aligned} & \Rightarrow a+(p-1) d+a+(q-1) d \\ & \quad=a+(r-1) d+a+(s-1) d \\ & \Rightarrow 2 a+(p+q) d-2 d=2 a+(r+s) d-2 d \\ & \Rightarrow(p+q) d=(r+s) d \Rightarrow p+q=r+s . \end{aligned} $$

Hence, the correct answer is option: 'D'.

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