Question

The variance of first $100$ odd natural numbers is

• A. $2222$
• B. $3333$
• C. $4444$
• D. $5555$

Answer 1

Answer: (B)

Variance $=\frac{1^{2} + 3^{2} + 5^{2} + \ldots . . + \left(199\right)^{2}}{100}-\left(\frac{1 + 3 + 5 + \ldots . . + 199}{100}\right)^{2}$ Sum $=\displaystyle \sum _{n = 1}^{n}\left(2 n - 1\right)^{2}=4\Sigma n^{2}-4\Sigma n+n=\frac{4 n \left(n + 1\right) \left(2 n + 1\right)}{6}-\frac{4 n \left(n + 1\right)}{2}+n$
Put $n=100$
$=\frac{4 \times 100 \times 101 \times 201}{6}-2\times 100\times 101+100$
$=100\left(\frac{2 \times 101 \times 201}{3} - 2 \times 101 + 1\right)=100\left(2 \times 6767 - 202 + 1\right)$
$=100\left(13534 - 201\right)=100\times 13333=1333300$
Variance $=\frac{1333300}{100}-\left(100\right)^{2}=13333-10000=3333$