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 ratio of the first term of the $A.P.$ to its common difference is

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

Answer 1

Answer: (D)

$-\frac{n}{2}$

Let $a$ and $d$ respectively be the first term and common difference of the $A.P.$

Given: $n=\frac{2mr}{m+r}$ ...(1)

and, ${(a+nd)}^2=(a+md)(a+rd)$

$\Rightarrow {(\frac{a}{d}+n)}^2=(\frac{a}{d}+m)(\frac{a}{d}+r)$

$\Rightarrow {(x+n)}^2=(x+m)(x+r)$ $[$ Putting $\frac{a}{d}=x]$

$\Rightarrow x^2+2nx+n^2=x^2+(m+r)x+mr$

$\Rightarrow (m+r-2n)x=n^2-mr=n^2-\frac{n(m+r)}{2}$ [Using (1)]

$=\frac{n}{2}(2n-m-r)$

$\therefore x=-\frac{n}{2}$ which is the required ratio.