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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 \frac{n^{2}-1}{4}

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

 
Answers (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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