Question

In a H.P, $t_n$ = $\frac{1}{n}$ . The Harmonic mean of first 10 terms of H.P is

• A. 5/11
• B. 2/11
• C. -5/11
• D. -2/11

Answer 1

The correct option is B

2/11

Given $t_n$ = $\frac{1}{n}$

$\Rightarrow$ 1, $\frac{1}{2}$, $\frac{1}{3}$,--------- forms H.P.

H.M. of n positive numbers $x_n$, $x_2$,----------- $x_n$ is,

H.M. = $\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+.......+\frac{1}{x_n}}$

Hence, H.M. of given H.P. = $\frac{10}{1+2+3+......+10}$

= $\frac{10\times 2}{10\times 11}$ [∑ n = $\frac{n(n+1)}{2}$ ]

= $\frac{2}{11}$

Short cut:- If $a_1$, $a_2$,----------- $a_n$ form H.P.

H.M. of all 'n' terms = $\frac{2a_1{a}_n}{a_1+a_n}$

Here $a_1$ = 1 $a_n$ = $\frac{1}{10}$ hence H M = $\frac{2\times \frac{1}{10}}{1+\frac{1}{10}}$ = $\frac{2}{11}$