Assertion (A): The variance of the first n od | vedclass

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(A) The sum of the first $n$ odd natural numbers is given by $\sum_{i=1}^{n} (2i-1) = n^2$.The sum of the squares of the first $n$ odd natural numbers

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$(A)$ and $(R)$ are true. $(R)$ is the correct explanation of $(A)$

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(A) The sum of the first $n$ odd natural numbers is given by $\sum_{i=1}^{n} (2i-1) = n^2$.The sum of the squares of the first $n$ odd natural numbers is $\sum_{i=1}^{n} (2i-1)^2 = \sum_{i=1}^{n} (4i^2 - 4i + 1) = 4 \sum i^2 - 4 \sum i + \sum 1$.Using standard summation formulas: $\sum i^2 = \frac{n(n+1)(2n+1)}{6}$ and $\sum i = \frac{n(n+1)}{2}$.Sum $= 4 \left[ \frac{n(n+1)(2n+1)}{6} \right] - 4 \left[ \frac{n(n+1)}{2} \right] + n = \frac{2n(n+1)(2n+1) - 6n(n+1) + 3n}{3} = \frac{n(4n^2-1)}{3}$.Thus,Reason $(R)$ is true.Variance is defined as $\sigma^2 = \frac{\sum x_i^2}{n} - \left( \frac{\sum x_i}{n} \right)^2$.$\sigma^2 = \frac{n(4n^2-1)}{3n} - \left( \frac{n^2}{n} \right)^2 = \frac{4n^2-1}{3} - n^2 = \frac{4n^2-1-3n^2}{3} = \frac{n^2-1}{3}$.Thus,Assertion $(A)$ is true and $(R)$ is the correct explanation of $(A)$.

	Physics	Chemistry	Mathematics	Biology
Mean	$20$	$25$	$23$	$27$
$S$.$D$.	$3$	$2$	$4$	$5$

Classes	$30-40$	$40-50$	$50-60$	$60-70$	$70-80$	$80-90$	$90-100$
Frequency $({f_i})$	$3$	$7$	$12$	$15$	$8$	$3$	$2$

Class	$10-20, 20-30, 30-40$
Frequency	$2, x, 2$

Assertion $(A)$: The variance of the first $n$ odd natural numbers is $\frac{n^2-1}{3}$.
Reason $(R)$: The sum of the first $n$ odd natural numbers is $n^2$ and the sum of the squares of the first $n$ odd natural numbers is $\frac{n(4n^2-1)}{3}$.
Which of the following alternatives is correct?

Similar Questions

The following table shows the information about marks obtained in Physics,Chemistry,Mathematics,and Biology by $100$ students in a class. Which subject shows the highest variability in marks?
Physics Chemistry Mathematics Biology
Mean $20$ $25$ $23$ $27$
$S$.$D$. $3$ $2$ $4$ $5$

Calculate the mean,variance,and standard deviation for the following distribution.

Classes $30-40$ $40-50$ $50-60$ $60-70$ $70-80$ $80-90$ $90-100$

Frequency $({f_i})$ $3$ $7$ $12$ $15$ $8$ $3$ $2$

In a series of $2n$ observations,half of them are equal to $a$ and the remaining half are equal to $-a$. If the standard deviation of the observations is $2$,then $|a|$ equals:

If the variance of the following frequency distribution is $50$,then $x$ is equal to:

Class $10-20, 20-30, 30-40$

Frequency $2, x, 2$

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Assertion $(A)$: The variance of the first $n$ odd natural numbers is $\frac{n^2-1}{3}$.Reason $(R)$: The sum of the first $n$ odd natural numbers is $n^2$ and the sum of the squares of the first $n$ odd natural numbers is $\frac{n(4n^2-1)}{3}$.Which of the following alternatives is correct?