Q: If ar > 0 , r ∈ N and a1 , a2 , a3 , …., a2n are in A.P, then
$\large \frac{a_1 + a_{2n}}{\sqrt{a_1} + \sqrt{a_2}} + \frac{a_2 + a_{2n-1}}{\sqrt{a_2} + \sqrt{a_3}} + …..+ \frac{a_n + a_{n+1}}{\sqrt{a_n} + \sqrt{a_{n+1}}}$ is equal to
(A) n-1
(B) $\large \frac{n (a_1 + a_{2n})}{\sqrt{a_1} + \sqrt{a_{n+1}}} $
(C) $\large \frac{n -1}{\sqrt{a_1} + \sqrt{a_{n+1}}} $
(D) none of these
Sol: (B) a1 + a2n = a2 + a2n–1 = ……… = an + an+1 = k (say)
now given expression $\large = k(\frac{\sqrt{a_1} – \sqrt{a_2}}{a_1-a_2} + …. + \frac{\sqrt{a_n} – \sqrt{a_{n+1}}}{a_n-a_{n+1}}) = -\frac{k}{d}(\sqrt{a_n} – \sqrt{a_{n+1}})$
$\large = -\frac{k}{d}\frac{a_1 – a_{n+1}}{\sqrt{a_1} + \sqrt{a_{n+1}}} $
$\large = (a_1 + a_{2n}) \frac{-nd}{-d(\sqrt{a_1} + \sqrt{a_{n+1}})} $
$\large = \frac{n (a_1 + a_{2n})}{\sqrt{a_1} + \sqrt{a_{n+1}}} $