Q

I have a doubt, kindly clarify. - Sequence and series - JEE Main-4

Directions : Questions are Assertion- Reason type questions. Each of these questions contains two statements :

Statement- 1 (Assertion) and Statement - 2 (Reason).

Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice.

Question :

Statement-1 : The variance of first $n$ even natural numbers is $\dpi{100} \frac{n^{2}-1}{4}$

Statement-2 : The sum of first $n$ natural numbers is $\dpi{100} \frac{n(n+1)}{2}$  and the sum of squares of first $n$ natural numbers is $\dpi{100} \frac{n(n+1)(2n+1)}{6}$

• Option 1)

Statement-1 is true,Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

• Option 2)

Statement- 1 is true, Statement-2 is false

• Option 3)

Statement-1 is false, Statement-2 is true

• Option 4)

Statement-1 is true, Statement-2 is true; Statement-2 is correct explanation for Statement-1

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As we learnt

Summation of series of natural numbers -

$\sum_{k=1}^{n}K= \frac{1}{2}n\left ( n+1 \right )$

- wherein

Sum of first n natural numbers

$1+2+3+4+------+n= \frac{n(n+1)}{2}$

Summation of series of natural numbers -

$\sum_{k=1}^{n}K^{2}= \frac{1}{6}n\left ( n+1 \right )\left ( 2n+1 \right )$

- wherein

Sum of  squares of first n natural numbers

$1^{2}+2^{2}+3^{2}+4^{2}+------+n^{2}= \frac{n(n+1)\left ( 2n+1 \right )}{6}$

$Variance=\frac{1}{n}\sum x_i^2-\overline{x}$

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

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

$=\frac{4}{n}\frac{n(n+1)(2n+1)}{6}-(n+1)$

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

$=\frac{\left ( n+1 \right )\left ( 4n-1 \right )}{3}$

Option 1)

Statement-1 is true,Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

Option 2)

Statement- 1 is true, Statement-2 is false

Option 3)

Statement-1 is false, Statement-2 is true

Option 4)

Statement-1 is true, Statement-2 is true; Statement-2 is correct explanation for Statement-1

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