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What is the value of C_{0}+3 C_{1}+5 C_{2}+\ldots+(2 n+1) C_{n}=?

Option: 1

(n+1) 2^{n}


Option: 2

(n-1) 2^{n}


Option: 3

n .2^{n}


Option: 4

2^{n}


Answers (1)

best_answer

 

 

Derivative of the Polynomial Function -

Derivative of the Polynomial Function

 

\\\mathbf{2.}\;\;\;\;\mathrm{\mathbf{\frac{\mathit{d}}{\mathit{dx}}(x^n)=nx^{n-1}}}

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x\left(1+x^{2}\right)^{n}=C_{0} x+C_{1} x^{3}+C_{2} x^{5}+\ldots+C_{n} x^{2 n+1}\\ {\text { On differentlating both sides w.r.t } x} \\ {x \cdot n\left(1+x^{2}\right)^{n-1} \cdot 2 x+\left(1+x^{2}\right)^{n} \cdot 1=C_{0}+3 C_{1} x^{2}+5 C_{2} x^{4}+\ldots+(2 n+1) C_{n} x^{2 n}} \\ {\text { Putting } x=1, \text { we get }} \\ n \cdot 2^{n-1} \cdot 2+2^{n}=C_{0}+3 C_{1}+5 C_{2}+\ldots+(2 n+1) C_{n}\\ C_{0}+3 C_{1}+5 C_{2}+\ldots+(2 n+1) C_{n}=(n+1) 2^{n}

Posted by

Irshad Anwar

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