Question

Between two quantities a and b are inserted n arithmetic means and also n harmonic means. If x and y be the $r^{th}$ terms of A.P. and H.P. respectively, then prove that $\frac{x}{a}+\frac{b}{y}$ is independent of r and n.

Answer 1

$A_n$ = nth A.M. between a and b, $A_n=\frac{nb+a}{n+1}$
$H_n$ = nth H.M. between a and b,
$H_n=\frac{(n+1)ab}{na+b}$
$T_r$ of A.P. is $A_{r-1}$ (mean) of A.P. $=x$
$T_r$ of H.P. is $H_{r-1}$ (mean) of H.P. $=y$
Hence putting n = r - 1 or n + 1 = r in (1) and (2), we get
$x=\frac{(r-1)b+a}{r},y=\frac{rab}{(r-1)a+b}$
$\therefore \frac{x}{a}+\frac{b}{y}=\left[\frac{(r-1)b+a}{ar}\right]+\left[\frac{(r-1)b+a}{ra}\right]$
$=\frac{(r-1)(a+b)+(a+b)}{ar}=\frac{r(a+b)}{ar}=\frac{a+b}{a}$
Hence in is independent of e and n.