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  • Find the value of <img alt="C_{0}+3 C_{1}+5 C_{2}+\ldots \ldots+(2 n+1) C_{n}" src="http://entrancecorner.oncodecogs.com/gif.latex?C_%7B0%7D&plus;3%20C_%7B1%7D&plus;5%20C_%7B2%7D&p

Find the value of C_{0}+3 C_{1}+5 C_{2}+\ldots \ldots+(2 n+1) C_{n}.

 If \begin{array}{l}{ \mathrm{C}_{0}, \mathrm{C}_{1}, \mathrm{C}_{2}, \ldots, \mathrm{C}_{n} }\end{array} be binomial coefficients in the expansion of (1+x)^n

Option: 1

(n+1) 2^{n}


Option: 2

(2n+1) 2^{n}


Option: 3

(2n-1) 2^{2n-1}


Option: 4

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


Answers (1)

best_answer

As we learnt

C_{1}+2 \cdot C_{2}+3 \cdot C_{3}+---+n \cdot C_{n}=\sum_{r=0}^{n} r \cdot{ }^{n} C_{r}=n \cdot 2^{n-1}

and

C_{0}+C_{1}+C_{2}+C_{3}+----+C_n= \sum_{r=0}^{n} (^{n} C_{r}) = 2^{n}

 

Now,

Given expression can be written as 

\sum_{r=0}^{n} (2r+1) ^{n} C_{r}

=2\sum_{r=0}^{n} (r) .^{n} C_{r} + \sum_{r=0}^{n} (^{n} C_{r})

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

 

Posted by

Sanket Gandhi

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