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}}"
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