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If x1 , x2, ..., xn are any real number...

If `x_1 , x_2, ..., x_n` are any real numbers and n is anypositive integer, then

A

`n underset(i=1)overset(n)sum x_(i)^(2) lt (underset(i=1)overset(n)sum x_(i))^(2)`

B

`n underset(i=1)overset(n)sum x_(i)^(2) ge (underset(i=1)overset(n)sum x_(i))^(2)`

C

`n underset(i=1)overset(n)sum x_(1)^(2) ge n (underset(i=1)overset(n)sum x_(i))^(2)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
Since, `x_(1) , x_(2),..,x_(n)` are positive real numbers.
`:.` Using nth power mean inequality
`(x_(1)^(2) + x_(2)^(2) + ...+ x_(n)^(2))/(n) ge ((x_(1) + x_(2) + ...+ x_(n))/(n))^(2)`
`rArr (n^(2))/(n) (underset(i=1)overset(n)sum x_(1)^(2)) ge (underset(i=1)overset(n)sum x_(i))^(2) rArr n (underset(i=1)overset(n)sumx_(i)^(2)) ge (underset(i=1)overset(n) sum x_(i))^(2)`
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