Question

The variance of first n natural numbers is ______.

Options

• `(n(n + 1))/2`

• `((n + 1)(n + 5))/12`

• `((n + 1)(n - 5))/12`

• `((n^2 - 1))/12`

Answer 1

The variance of first n natural numbers is `underlinebb(((n^2 - 1))/12)`.

Explanation:

Variance of first n natural numbers

= `(sumn^2)/n - ((sumn)/n)^2`

= `(n(n + 1)(2n + 1))/(6n) - ((n(n + 1))/(2n))^2`

= `(n + 1)[(2n + 1)/6 - ((n + 1))/4]`

= `((n + 1))/12[4n + 2 - 3n - 3]`

= `((n + 1))/12 xx (n - 1)`

= `(n^2 - 1)/12`