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)

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