Evaluate:
$1^{2} c_{1} + 2^{2} c_{2} + 3^{2} c_{3} + 4^{2} c_{4} + . . . . . . + n^{2} c_{n}$

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Solution

$s = 1^{2} C_{1} + 2^{2} C_{2} + 3^{2} C_{3} + 4^{2} C_{4} + . . . . . + n^{2} C_{n}$

$(1 + x)^{n} =^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} +^{n} C_{3} x^{3} + . . . . . +^{n} C_{n} x^{n}$

Differentiable w.r.t. $x$

$n (1 + x)^{n - 1} = 0 +^{n} C_{1} + 2^{n} C_{2} x + 3^{n} C_{3} x^{2} + . . . . . + n^{n} C_{x} x^{n - 1}$

Multiply whole expression with x

$n x (1 + x)^{n - 1} =^{n} C_{1} x + 2^{n} C_{2} x^{2} + 3^{n} C_{3} x^{3} + . . . \dots + n^{n} C_{n} x^{n}$

Differentiable w.r.t. $x$ again

$n (1 + x)^{n - 1} + n x (n - 1) (1 + x)^{n - 2} =^{n} C_{1} + 2^{2}^{n} C_{2} x + 3^{2}^{n} C_{3} x^{2} + n^{n} C_{n} x^{n - 1}$

Put $x = 1$

$\Rightarrow S =^{n} C_{1} + 2^{2}^{n} C_{2} + 3^{2}^{n} C_{3} + . . . \dots . + n^{2}^{n} C_{n}$

$= n (1 + 1)^{n - 1} + n .1 . (n - 1) (1 + 1)^{n - 2}$

$= n 2^{n - 1} + n (n - 1) 2^{n - 2}$

$= n 2^{n - 2} (2 + (n - 1)) = n 2^{n - 2} (n + 1)$ .

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Similar questions

(1 + x)^{n} = c_{0} + c_{1} x + c_{2} x^{2} + . . . . + c_{n} x^{n}

1^{2} c_{1} + 2^{2} c_{2} + 3^{2} c_{3} + . . . . . + n^{2} c_{n}

C_{0}, C_{1}, C_{2} . . . C_{n}

{(1 + x)}^{n}

1^{2} . C_{1} + 2^{2} C_{2} + 3^{2} C_{3} + . . . + n^{2} C_{n}

x^{49}

(x - \frac{C_{1}}{C_{0}}) (x - 2^{2} \frac{C_{2}}{C_{1}}) (x - 3^{2} \frac{C_{3}}{C_{2}}) . . . . . (x - 50^{2} \frac{C_{50}}{C_{49}})

With usual notations, $C_{1} + 2^{2} C_{2} + 3^{2} C_{3} + . . . . . . . =$

(1 + x)^{n} = c_{0} + c_{1} x + c_{2} x^{2} + c_{3} x^{3} + . . . . + c_{n} x^{n},

(n - 1)^{2} c_{1} + (n - 3)^{2} c_{3} + (n - 5)^{2} c_{5} + . . . . .

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