JEE Main 2017 · Mathematics · Statistics · medium

The variance of the first n odd natural numbers 1,3,5,…,(2n-1) is 16. Then n equals

Single correct option

The variance of the first

n

odd natural numbers

1,3,5,\dots,(2n-1)

16

. Then

n

equals

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Answer: (B) $7$

The numbers are

2k-1

for

k=1,\dots,n

, an affine function of

1,2,\dots,n

. Variance is unaffected by shifting and scales by the square of the multiplier, so it equals

4\times\mathrm{Var}(1,\dots,n)=4\cdot\dfrac{n^2-1}{12}=\dfrac{n^2-1}{3}

. Setting this to

16

gives

n^2=49

, so

n=7

. Check: for

1,3,\dots,13

the mean is

7

and the mean of squares is

\tfrac{455}{7}=65

, giving variance

65-49=16

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