Question

Find the H.M. of 1/2,2/3,3/4,.......n/(n+1) occurring with frequencies 1,2,3,.....n

Answer 1

The harmonic mean can be found as

$$HM = {{\sum\limits_{k = 1}^n {{f_k}} } \over {\sum\limits_{k = 1}^n {{{{f_k}} \over {{x_k}}}} }}$$

where ${{x_k}}$ are dara values and ${{f_k}}$ are corresponding frequencies. We've got ${f_k} = k$ and ${x_k} = {k \over {k + 1}}$ . Thus we have to find two sums. The first is just the arithmetic progression

$$\sum\limits_{k = 1}^n {{f_k}} = \sum\limits_{k = 1}^n k = {{n(n + 1)} \over 2}$$

The second can be simplifid to the arithmetic progression and the sum of units

$$\sum\limits_{k = 1}^n {{{{f_k}} \over {{x_k}}}} = \sum\limits_{k = 1}^n {{k \over {{k \over {k + 1}}}}} = \sum\limits_{k = 1}^n {(k + 1)} = \sum\limits_{k = 1}^n k + \sum\limits_{k = 1}^n 1 = {{n(n + 1)} \over 2} + n$$

Then

$$HM = {{{{n(n + 1)} \over 2}} \over {{{n(n + 1)} \over 2} + n}} = {2 \over {n(n + 1) + 2n}} \cdot {{n(n + 1)} \over 2} = {{n + 1} \over {n + 3}}$$