Find the harmonic mean of 12,23,34,…,nn+1 occurring with frequencies 1, 2, 3, ..., n, respectively.

We know that, Harmonic mean =∑f∑fx∴ ∑f=1+2+3+…+n=n(n+1)2 and  ∑fx=11/2+22/3+33/4+…+nn/(n+1)=2+3×22+4×33+…+n(n+1)n=2+3+4+…+n+(n+1)which is an arithmetic progression with a.=2 and d =1. By the formula of sum of n term of an AP,∑fx=n2{2a+(n−1)d}, we have =n2{2×2+n−1}=n2(3+n) Harmonicmean=n(n+1)÷2 n(3+n)÷2=n+13+n