$\sqrt{x_{1}-1^{2}} + 2 \sqrt{x_{2}-2^{2}} + \ldots + n \sqrt{x_{n}-n^{2}} = \frac{1}{2} \left [ x_{1} + x_{2}+ \ldots + x_{n} \right ]$

Then which of the following is true?

$x_{n}=n$
$x_{n}=2n^{2}$
$x_{n}=2n$
Cannot be determined
$x_{n}=n^{2}$

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each term on the LHS containing x1 , x2, x3 ......should finally yield (x1/2 ),(x2/2) ,(x3 /2),..... as each term is unique and there is only one source(term) from which a particular variable comes, so Xn/2=n[(Xn - n^2)^1/2] solving this we get Xn=2n^2