Question

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

Answer 1

$m,n,r$ are in H.P.

Therefore $\frac{1}{m}+\frac{1}{r}=\frac{2}{n}$

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

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

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

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

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