Question

If $a_1,a_2,a_3\dots a_n$ are in H.P. and $f(k)=\left({\sum }_{r=1}^n{a}_r\right)-a_k$, then $\frac{a_1}{f(1)},\frac{a_2}{f(2)},\frac{a_3}{f(3)},\dots ,\frac{a_n}{f(n)}$ are in

• A. A.P.
• B. GP.
• C. H.P.
• D. None of these

Answer 1

Answer: (C)

$f(k)+a_k=\sum _{r=1}{a}_r=\lambda$ (say)
$\therefore f(k)=\lambda -a_k$
$\Rightarrow \frac{f(k)}{a_k}=\frac{{\lambda }^{r=1}}{a_k}-1$
$\therefore \frac{f(1)}{a_1},\frac{f(2)}{a_2},\dots ,\frac{f(n)}{a_n}$ are in A.P.
So $\frac{a_1}{f(1)},\frac{a_2}{f(2)},\dots ,\frac{a_n}{f(n)}$ are in H.P.