Question

The harmonic mean of sum of first n natural numbers and sum of cubes of first n natural numbers is

• A. n²(n+1)² / (n² + n + 2)
• B. n²(n+1)² / (n² + n + 1)
• C. n²(n+1)² / (n² + 2n+ 3 )
• D. n²(n+1)² / (n² + n + 4 )

Answer 1

The correct option is A

n²(n+1)² / (n² + n + 2)

Sum of first n natural numbers = $\frac{n(n+1)}{2}$

Sum of cubes of first n natural numbers = $\frac{n^2{(n+1)}^2}{4}$

Harmonic mean = $\frac{\left(2^{\ast }{\left[\frac{n(n+1)}{2}\right]}^{\ast }\left[\frac{n^2(n+1{)}^2}{4}\right]\right)}{(\left[\frac{n(n+1)}{2}\right]+\left[\frac{n^2(n+1{)}^2}{4}\right])}$

$=\frac{n^2(n+1{)}^2}{(n^2+n+2)}$