Sequence And Series Question 226
Question: If the $ {{(m+1)}^{th}},\ {{(n+1)}^{th}} $ and $ {{(r+1)}^{th}} $ terms of an A.P. are in G.P. and $ m,\ n,\ r $ are in H.P., then the value of the ratio of the common difference to the first term of the A.P. is
[MNR 1989; Roorkee 1994]
Options:
A) $ -\frac{2}{n} $
B) $ \frac{2}{n} $
C) $ -\frac{n}{2} $
D) $ \frac{n}{2} $
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ a $ be the first term and d be the common difference of the given A.P. Then as given the $ {{(m+1)}^{th}} $ , $ {{(n+1)}^{th}} $ and $ {{(r+1)}^{th}} $ terms are in G.P.
$ \Rightarrow $ $ a+md,\ a+nd,\ a+rd $ are in G.P.
$ \Rightarrow $ $ {{(a+nd)}^{2}}=(a+md)(a+rd) $
$ \Rightarrow $ $ a(2n-m-r)=d(mr-n^{2}) $ or $ \frac{d}{a}=\frac{2n-(m+r)}{mr-n^{2}} $ ?..(i) Next, $ m,\ n,\ r $ in H.P.
$ \Rightarrow n=\frac{2mr}{m+r} $ ?..(ii) From (i) and (ii) $ \frac{d}{a}=\frac{2n-(m+r)}{mr-n^{2}}=\frac{2}{n}( \frac{2n-(m+r)}{(m+r)-2n} )=-\frac{2}{n} $ .
BETA
