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Assertion (A): The variance of the first $\mathrm{n}$ odd natural numbers is $\frac{n^2-1}{3}$.
Reason (R): The sum of the first $\mathrm{n}$ odd natural numbers is $\mathrm{n}^2$ and the sum of the squares of the first $\mathrm{n}$ odd natural numbers is $\frac{n\left(4 n^2-1\right)}{3}$.
Which of the following alternatives is correct?
Options:
Reason (R): The sum of the first $\mathrm{n}$ odd natural numbers is $\mathrm{n}^2$ and the sum of the squares of the first $\mathrm{n}$ odd natural numbers is $\frac{n\left(4 n^2-1\right)}{3}$.
Which of the following alternatives is correct?
Solution:
1816 Upvotes
Verified Answer
The correct answer is:
(A) and (R) are true. (R) is correct explanation of (A)
Sum of first $n$ odd natural numbers
$$
1+3+5+\ldots . .+(2 n-1)=\frac{n}{2}(1+2 n-1)=n^2
$$
Sum of squares of first $n$ odd natural numbers
$$
\begin{aligned}
& =\sum(2 n-1)^2=\sum\left(4 n^2+1-4 n\right) \\
& =\frac{4 n(n+1)(2 n+1)}{6}-\frac{4 n(n+1)}{2}+n \\
& =\frac{2 n(n+1)(2 n+1)-6 n(n+1)+n}{3} \\
& =\frac{n}{3}\left(4 n^2+6 n+2-6 n-6+3\right)
\end{aligned}
$$
$$
\begin{aligned}
& =\frac{n}{3}\left(4 n^2-1\right) \\
& \text { Variance }=\frac{\Sigma x_i^2}{n}-\left(\frac{\Sigma x_i}{n}\right)^2 \\
& =\frac{n}{3 n}\left(4 n^2-1\right)-\left(\frac{n^2}{n}\right)^2 \\
& =\frac{4 n^2-1}{3}-n^2=\frac{n^2-1}{3}
\end{aligned}
$$
$\therefore A$ and $R$ are true. $R$ is correct explanation of $A$.
$$
1+3+5+\ldots . .+(2 n-1)=\frac{n}{2}(1+2 n-1)=n^2
$$
Sum of squares of first $n$ odd natural numbers
$$
\begin{aligned}
& =\sum(2 n-1)^2=\sum\left(4 n^2+1-4 n\right) \\
& =\frac{4 n(n+1)(2 n+1)}{6}-\frac{4 n(n+1)}{2}+n \\
& =\frac{2 n(n+1)(2 n+1)-6 n(n+1)+n}{3} \\
& =\frac{n}{3}\left(4 n^2+6 n+2-6 n-6+3\right)
\end{aligned}
$$
$$
\begin{aligned}
& =\frac{n}{3}\left(4 n^2-1\right) \\
& \text { Variance }=\frac{\Sigma x_i^2}{n}-\left(\frac{\Sigma x_i}{n}\right)^2 \\
& =\frac{n}{3 n}\left(4 n^2-1\right)-\left(\frac{n^2}{n}\right)^2 \\
& =\frac{4 n^2-1}{3}-n^2=\frac{n^2-1}{3}
\end{aligned}
$$
$\therefore A$ and $R$ are true. $R$ is correct explanation of $A$.
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