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MHT CET202415 May 2024Evening ShiftMathematicsStatisticsActual

Variance of first n natural numbers is .

Options

  1. An ^2- 1 12
  2. B( n -1)^2 12
  3. Cn ^2 12 -1
  4. Dn ^2-1 12

Correct answer

D. n ^2-1 12

Step-by-step solution

aligned & ^2= 1 n (1^2+2^2+3^2+ +n^2 ) &- ( 1+2+3+ +n n )^2 aligned aligned & = 1 n ( n(n+1)(2 n+1) 6 )- ( n(n+1) 2 n )^2 & = (n+1)(2 n+1) 6 - ( n+1 2 )^2 & = n+1 2 ( 2 n+1 3 - n+1 2 ) & = n+1 2 ( n-1 6 ) & = n^2-1 12 aligned

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