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The mean of the series a, a + d, a + 2d,...

The mean of the series a, a + d, a + 2d, …, a + 2 nd, is

A

`(n(n+1))/2d^2`

B

`(n(n+1))/3d^2`

C

`(n(n+1))/6d^2`

D

`(n(n+1))/12d^2`

Text Solution

Verified by Experts

The correct Answer is:
B
Given series a,a+d,a+2d,…., a+2nd
`barx=(((2n+1)/2)(a+a+2nd))/(2n+1)=a+nd`
`Sigma(x_i-barx)^2=n^2 d^2 + (1-n)^2 d^2` +….+ `d^2+0+d^2`+….+`n^2d^2`
`=2d^2 (n^2+(n-1)^2+….+1^2)=(n(n+1)(2n+1))/3 d^2`
`therefore` Variance =`(Sigma(x_i-barx)^2)/(2n+1)=(n(n+1))/3 d^2`
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