Question

If $(m+1{)}^{th},(n+1{)}^{th}\text{and}(r+1{)}^{th}$ terms of an A.P. are in G.P and $m,n,r$ are in H.P., then ratio of the first term of the A.P. to its common difference in terms of $n$ is

• A. $\frac{n}{2}$
• B. $-\frac{n}{2}$
• C. $\frac{n}{3}$
• D. $-\frac{n}{3}$

Answer 1

The correct option is B $-\frac{n}{2}$

$a_{m+1}=a+\left(m\underset{-1}{+1}\right)d$

$=a+md$

$a_{n+1}=a+(n+1-1)d=a+nd$

$a_{r+1}=a+rd$

These three terms are in G.P.

$\therefore (a+nd{)}^2=(a+md\left)\right(a+rd)$

$a^2+n^2{d}^2+2and=a^2+amd+ard+mr{d}^2$

$d^2(n^2-mr)=d(am+ar+2an)$

$d^2(n^2-mr)=ad(m+r-2n)$

$\Rightarrow n=\frac{2mr}{m+r}$ [$\because n,m,r$ are in $H{P}^{-}$]

$m+r=\frac{n}{2mr}$

Substituting

$\frac{a}{d}=\frac{n^2-mr}{\frac{2mr}{n}-2n}={\frac{(n^2-mr)}{\frac{2}{n}(mr-n^2)}}^{-1}$

$\frac{a}{d}=-\frac{n}{2}$