Question

If the $( m +1)^{ th },( n +1)^{ th } \&( r +1)^{ th }$ terms of an AP are in GP & $m , n , r$ are in HP, then the ratio of the common difference to the first term of the AP is -

• A. $\frac{1}{ n }$
• B. $\frac{2}{n}$
• C. $-\frac{2}{n}$
• D. none of these

Answer 1

Answer: (C)

$\Rightarrow \frac{ a + rd }{ a + nd }=\frac{ a + nd }{ a + md } $
$\Rightarrow \frac{1+ r ( d / a )}{1+ n ( d / a )}=\frac{1+ n ( d / a )}{1+ m ( d / a )} \text { Let } \frac{ d }{ a }= x$
$\Rightarrow (1+ nx )^2=(1+ rx )(1+ mx )$
$\Rightarrow \left( n ^2- mr \right) x ^2+(2 n - r - m ) x =0 \Rightarrow x =0$
or $x=-\left(\frac{2 n-r-m}{n^2-m r}\right)=\left(\frac{2 n-r-m}{\frac{n(m+r)}{2}-n^2}\right)=\frac{-2}{n}$
$\left( m , n , r \text { are in H.P. } \Rightarrow mr =\frac{ n ( m + r )}{2}\right)$