99 Percentile Qs Bank for JEE MainMathematicsStatistics
Assertion (A): The variance of the first n odd natural numbers is 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 n (4 n^2-1 ) 3 . Which of the following alternatives is correct?
Options
- A(A) and (R) are true. (R) is correct explanation of (A)
- B(A) and (R) are true, but (R) is not a correct explanation of ( A )
- C(A) is true, but (R) is false
- D(A) is false, but (R) is true
Correct answer
A. (A) and (R) are true. (R) is correct explanation of (A)
Step-by-step solution
Sum of first n odd natural numbers 1+3+5+ . .+(2 n-1)= n 2 (1+2 n-1)=n^2 Sum of squares of first n odd natural numbers aligned & = (2 n-1)^2= (4 n^2+1-4 n ) & = 4 n(n+1)(2 n+1) 6 - 4 n(n+1) 2 +n & = 2 n(n+1)(2 n+1)-6 n(n+1)+n 3 & = n 3 (4 n^2+6 n+2-6 n-6+3 ) aligned aligned & = n 3 (4 n^2-1 ) & Variance = x_i^2 n - ( x_i n )^2 & = n 3 n (4 n^2-1 )- ( n^2 n )^2 & = 4 n^2-1 3 -n^2= n^2-1 3 aligned A and R are true. R is correct explanation of A .