Combinatorial binomial

Question

Calculate the sum value: $$ \sum_ {k = 0} ^ {n} k \cdot 3 ^ {k} \cdot \binom {n} {k} $$ Tip: $$ \binom {n} {k} = \binom {n-1} {k-1} \cdot \frac {n} {k} $$ I can get here: $$\sum_ {k = 0} ^ {n} 3 ^ {k} \cdot \binom {n-1} {k-1} \cdot n $$ After that, the original resolution takes 'n' out of the sum and places it outside in the '3n' form. I have no idea where this three came from. Then they rewrite the expression this way: $$3n \cdot \sum_ {k = 1} ^ {n} 3 ^ {k} \binom {n-1} {k-1} $$$$ 3n \cdot (3 + 1) ^ {n-1} $$$$ 3n \cdot 4 ^ {n-1} $$ Could someone simply elude the above steps or develop a workaround? Original source of the problem: https://portaldosaber.obmep.org.br/uploads/material/dgr8vm27u08os.pdf

Answer 1

We obtain \begin{align*} \color{blue}{\sum_{k=0}^nk\,3^k\binom{n}{k}}&=\sum_{k=1}^nk\,3^k\binom{n}{k}\tag{1}\\ &=\sum_{k=1}^nk\,3^k\frac{n}{k}\binom{n-1}{k-1}\tag{2}\\ &=n\sum_{k=1}^n3^k\binom{n-1}{k-1}\tag{3}\\ &=n\sum_{k=0}^{n-1}3^{k+1}\binom{n-1}{k}\tag{4}\\ &=3n\sum_{k=0}^{n-1}3^{k}\binom{n-1}{k}\tag{5}\\ &\,\,\color{blue}{=3n4^{n-1}}\tag{6}\\ \end{align*} and the claim follows.

Comment:

• In (1) we skip the index $k=0$ which does not contribute anything.

• In (2) we apply the binomial identity $\binom{n}{k}=\frac{n}{k}\binom{n-1}{k-1}$.

• In (3) we cancel $k$ and factor out $n$.

• In (4) we shift the index and start with $k=0$. We compensate the shift of the index $k$ by replacing in the summand each occurrence of $k$ with $k+1$.

• In (5) we factor out $3$.

• In (6) we apply the binomial theorem to $4^{n-1}=(1+3)^{n-1}$.

Answer 2

You are right, but another way is: $$(1+x)^{n}=\sum_{k=0}^{n} {n \choose k} x^k$$, D.w.r.t.x we get $$n(1+x)^{n-1}=\sum_{k=0}^{n} k {n \choose k} x^{k-1}$$ By putting $x=3$, we get the sum as $3n 4^{n-1}$