$$\sum_{k=0}^{n} 3^k \binom{n}{k}$$
I have tried filling in and simplifying but the following was not correct: $$1 + 3n + 3^2 \dbinom{n}{2} + \cdots + 3^n$$
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4It is just $4^n$, by the binomial theorem applied to $(3+1)^n$. – Jack D'Aurizio Oct 04 '15 at 20:37
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Why are we using (3+1) – Anonymous Oct 04 '15 at 20:52
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1Take $a=3$ and $b=1$ in $(a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^k b^{n-k}.$ – Jack D'Aurizio Oct 04 '15 at 20:52