Question

If $l,m,n$ are three numbers in G.P., prove that the first term of an A.P. whose $l^{th},m^{th},$ and $n^{th}$ terms are in H.P. is to be common difference as $m+1$ to $1$.

Answer 1

If $x,y,z$ are in H.P. then $\frac{x}{z}=\frac{x-y}{y-z}$

$l,m,n$ are in G.P. Therefore $\frac{l-m}{m-n}=\frac{l}{m}$ ....(1)

We have $a+(l-1)d,a+(m-1)d,a+(n-1)d$ in H.P.

Therefore, $\frac{a+(l-1)d}{a+(n-1)d}=\frac{(l-m)d}{(m-n)d}=\frac{l-m}{m-n}=\frac{l}{m}$ ....$\text{[From (1)]}$

$\Rightarrow a(l-m)=d\left(l\right(m-n)+(l-m\left)\right)$

$\Rightarrow \frac{a}{d}=\frac{l(m-n)}{l-m}+1=m+1$