Conclude the value of the sum: S_n = ((n),(0) ) + 2 ((n),(1) )+...+ 2^n ((n),(n) ) with help of (1+2x)^n


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Question Number 178530 by Acem last updated on 17/Oct/22

$C o n c l u d e t h e v a l u e o f t h e s u m :$ $S_{n} = (\begin{matrix} n \\ 0 \end{matrix}) + 2 (\begin{matrix} n \\ 1 \end{matrix}) + . . . + 2^{n} (\begin{matrix} n \\ n \end{matrix})$ $w i t h h e l p o f {(1 + 2 x)}^{n}$
	Answered by Ar Brandon last updated on 17/Oct/22

	$S = \underset{k = 0}{\sum^{n}} \overset{n}{} C_{k} 2^{k} = {(2 + 1)}^{n} = 3^{n}$ $\underset{k = 0}{\sum^{n}} \overset{n}{} C_{k} a^{k} b^{n - k} = {(a + b)}^{n}$
	Commented by Acem last updated on 17/Oct/22

	$G r e a t S i r!$
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