Let n be an odd natural number such that the variance of 1,2,3,4,,n is 14.

Question

Let n be an odd natural number such that the variance of $1,2,3,4,\ldots,n$ is 14. Then n is equal to

Answer: 13

Accepted Answer

$$14 = \frac{n(n+1)(2n+1)}{6n}-\left(\frac{n+1}{2}\right)^2 = \frac{(n+1)(2n+1)}{6}-\frac{(n+1)^2}{4}$$

$$14 = \frac{n+1}{12}[2(2n+1)-3(n+1)] = \frac{(n+1)(n-1)}{12}$$

$$168 = n^2-1 \implies n^2=169 \implies n=13$$

Solutions · 1