"with n a positive integer, evaluate the sum" - bionomial quesiton

Question

The question is, with n a positive integer, evaluate the sum

$c(n,0) + 2*c(n,1) + 2^2*c(n,2) + ... + 2^k*(n,k) + ... + 2^n(n,n)$

I think since $(a+b)^n = \sum_{k=0}^n c(n,k)b^ka^{n-k}$, first thing I should do is identify the values of $a$ and $b$ So since $c(n,0)2^0 + c(n,1)2^1*1^n-k + c(n,2)2^2*1^n-2 ..... ^n (n,n)$, i figured out $a = 1$ and $b = 2$ so answer is $3n$ (i just guessed the numbers. Is there any other way than guessing?)

Am I correct? step by step explanation would be very much appreciated. (and Im really sorry for the latex because Im a newbie so im unsure of how to do latex)

Answer 1

I think you are trying to evaluate

$$\sum_{k=0}^n \binom{n}{k}2^k.$$

Just observe that

$$\sum_{k=0}^n \binom{n}{k}2^k=\sum_{k=0}^n \binom{n}{k}2^k1^{n-k}=(2+1)^n=3^n.$$

The first equality comes from the fact that $2^k=2^k\cdot 1=2^k\cdot 1^{n-k}$.

The second to the last equality is just the binomial theorem.