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Find

$\sum_{r=0}^n3^r\binom{n}r$

When I tried to do it I got

$1 + 3n + 3^2\cdot \frac{n(n-1)}2 + \ldots + 3^{n-1}\cdot n + 3^n$

but I couldn't equate it to $4^n$ which is the answer I got by substituting $n$ for various numbers.

Can you tell me the correct mathematical way of getting the answer?

  • 0
    I took the liberty of changing your $_nC_r$ notation to the preferable $\binom{n}r$.2012-12-09

1 Answers 1

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Just use the binomial theorem. It says that

$(x+y)^n=\sum_{r=0}^n\binom{n}rx^ry^{n-r}\;;\tag{1}$

now let $x=3$ and $y=1$, and see that the righthand side of $(1)$ is exactly your sum, while the lefthand side is indeed $4^n$.