By using calculus on $(1+x)^n$ and setting x equal to some appropriate number, evaluate:

Question

$\sum_{r=1}^{n}{n \choose r} 3^rr$

This has been really bugging me as to the method involved in solving this equation.

Answer 1

Start with:

$$(1+x)^n=\sum_{r=0}^n\binom nrx^r$$

Differentiate both sides to get:

$$n(1+x)^{n-1}=\sum_{r=1}^n\binom nrrx^{r-1}$$

Multiply by $x$ to get:

$$xn(1+x)^{n-1}=\sum_{r=1}^n\binom nrrx^r$$

Finally, setting $x=3$ the sum evaluates to $3n\cdot4^{n-1}$