Question

If $a_{1}, a_{2}, a_{3} \ldots a_{n}$ are in H.P. and $f(k)=\left(\sum_{r=1}^{n} a_{r}\right)-a_{k}$, then $\frac{a_{1}}{f(1)}, \frac{a_{2}}{f(2)}, \frac{a_{3}}{f(3)}, \ldots, \frac{a_{n}}{f(n)}$ are in

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

Answer 1

Answer: (C)

$f(k)+a_{k}=\displaystyle\sum_{r=1} a_{r}=\lambda$ (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}}, \ldots, \frac{f(n)}{a_{n}}$ are in A.P.
So $\frac{a_{1}}{f(1)}, \frac{a_{2}}{f(2)}, \ldots, \frac{a_{n}}{f(n)}$ are in H.P.