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  • If <img alt="(1+x)^{15}=C_0+C_1 x+C_2 x^2+\ldots \ldots+C_{15} x^{15}" src="https://entrancec

If (1+x)^{15}=C_0+C_1 x+C_2 x^2+\ldots \ldots+C_{15} x^{15}, then C_2+2 C_3+3 C_4+\ldots .+14 C_{15}=

Option: 1

14.2^{14}


Option: 2

13.2^{14}+1


Option: 3

13.2^{14}-1


Option: 4

None of these

 


Answers (1)

best_answer

We have 

(1+x)^{15}=C_0+C_1 x+C_2 x^2,+\ldots .+C_{15} x^{15}

\Rightarrow \quad \frac{(1+x)^{15}-1}{x}=C_1+C_2 x+\ldots+C_{15} x^{14}

Differentiating both sides with respect to x,  we get

=\frac{x \cdot 15(1+x)^{14}-(1+x)^{15}+1}{x^2}

=C_2+2 C_3 x+\ldots \ldots+{ }^{14} C_{15} x^{13}

Putting , x=1 we get

C_2+2 C_3+\ldots+14 C_{15}=15 \cdot 2^{14}-2^{15}+1=13 \cdot 2^{14}+1

 

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Sayak

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