QJEEM20Ma94c8: JEE Main 2020 Mathematics - Class 11 JEEM20Ma94c8 Solution: If the variance of the first n natural numbers is 10 and the…

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If the variance of the first n natural numbers is 10 and the variance of the first m even natural numbers is 16, then m + n is equal to_____.

Accepted Answer

Variance

σ^{2} = \frac{\sum x_{i}^{2}}{N} - μ^{2}

variance of (1, 2, ….. n)

10 =

\frac{1^{2} + 2^{2} + . . . . + n^{2}}{n} - {(\frac{1 + 2 + 3 + . . . . + n}{n})}^{2}

on solving we get n = 11

variance of 2, 4, 6…….2m = 16

\Rightarrow

\frac{2^{2} + 4^{2} + . . . . + {(2 m)}^{2}}{m} - {(m + 1)}^{2}

= 16

\Rightarrow

m² = 49

\Rightarrow

m = 7

∴

m + n = 18

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