Question

If $(m+1{)}^{th},(n+1{)}^{th}$ 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. $-\frac{n}{2}$
• B. $-\frac{m}{2}$
• C. $r$
• D. $-\frac{mr}{m+r}$

Answer 1

Answer: (A), (D)

The correct options are
A $-\frac{n}{2}$
D $-\frac{mr}{m+r}$

Given $(a+nd{)}^2=(a+md\left)\right(a+rd)$
$\Rightarrow (\frac{a}{d}+n{)}^2=(\frac{a}{d}+m\left)\right(\frac{a}{d}+r)....(i)$
Also $n=\frac{2mr}{m+r}\Rightarrow mr=\frac{(m+r)n}{2}.....\left(ii\right)$
Now from (i), $(\frac{a}{d}{)}^2+2(\frac{an}{d})+n^2=(\frac{a}{d}{)}^2+(m+r)\frac{a}{d}+mr$
$\Rightarrow \frac{a}{d}=\frac{n^2-mr}{m+r-2n}=\frac{n^2-\frac{(m+r)n}{2}}{m+r-2n}$ from (ii)
$\therefore \frac{a}{d}=-\frac{n}{2}=-\frac{mr}{m+r}$