C_0+3 C_1+5 C_2+ +(2 n+1) C_n =? - Vidyadip

MCQ

$\left\{C_0+3 C_1+5 C_2+\ldots+(2 n+1) C_n\right\}=?$

A
$( n -2) . 2^{ n -2}$
B
$( n -1)( n +2)$
C
$( n +2) . 2^{ n -1}$
D
$(n+1) 2^n$

✓

Answer

(d) $( n +1) 2^{ n }$
Explanation: We have, $C _0+3 C _1+5 C _2+\ldots+(2 n +1) C _{ n }$
$\begin{array}{l}=\left(C_0+C_1+C_2+\ldots+C_n\right)+2\left(C_1+2 C_2+\ldots+n C_n\right) \\
=2^n+2\left(n \cdot 2^{n-1}\right)=(n+1) \cdot 2^n\end{array}$

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