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 in terms of $n$ be $n/k$ .Find $-k$ ?

Answer 1

$m,n,r$ are in H.P. implies $n=\frac{2mr}{m+r}$
$(m+1)$th , $(n+1)$th and $(r+1)$th terms of an A.P are in G.P implies
$(a+md)\times (a+rd)=(a+nd{)}^2$
$\Rightarrow a^2+ard+amd+mr{d}^2=a^2+2and+n^2{d}^2$
$\Rightarrow d^2(n^2-mr)=(am+ar-2an)d$
So, $\frac{a}{d}=\frac{n^2-mr}{m+r-2n}$
From the first condition, $m+r=\frac{2mr}{n}$
So, $\frac{a}{d}=\frac{n^3-mnr}{2mr-2n^2}=\frac{-n}{2}$
Thus, we get $k$ to be $-2$