$C_{0} - 2 C_{1} + 3 C_{2} - 4 C_{3} + . . . . + (- 1)^{n} (n + 1) C_{n} = 0$ .

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Solution

$(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + C_{3} x^{3} + . . . . . . . . + C_{n} x^{n}$
$(1 + x)^{n} \times x = C_{0} x + C_{1} x^{2} + C_{2} x^{3} + C_{3} x^{4} + . . . . . . . + C_{n} x^{n + 1}$
Differentisting both sides:
$(1 + x)^{n} + n x (1 + x)^{n - 1} = C_{0} + 2 C_{1} x + 3 C_{2} x^{2} + 4 C_{3} x^{3} + . . . . . . + (n + 1) C_{n} x^{n}$
Putting $x = - 1$
$\Rightarrow O = C_{0} - 2 C_{1} + 3 C_{2} - 4 C_{3} . . . . . . + (- 1)^{x} (n + 1) C_{n}$ Hence proved

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