First slide

Binomial theorem for positive integral Index

Question

If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n}$ them the value of $C_{0} + 2 C_{1} + 3 C_{2} + \dots + (n + 1) C_{n}$ will be

Moderate

A

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

$(n + 1) 2^{n}$
C

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

$(n + 2) 2^{n}$

Solution

Since, $x (1 + x)^{n} = {xC}_{0} + C_{1} x^{2} + C_{2} x^{3} + \dots + C_{n} x^{n + 1}$
On differentiating w.r.t. x, we get
$(1 + x)^{n} + nx (1 + x)^{n - 1} = C_{0} + 2 C_{1} x + 3 C_{2} x^{2}$ $+ \dots + (n + 1) C_{n} x^{n}$
Put x = L, we get $C_{0} + 2 C_{1} + 3 C_{2} + \dots + (n + 1) C_{n}$
$= 2^{n} + n 2^{n - 1} = 2^{n - 1} (n + 2)$

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