Induction proof on a summation

Question

I'm trying to do an induction problem but I have little experience with summations and need some assistance, the problem is:

$$\sum_{r = 0}^{n} {{n}\choose{r}}6^r = 7^n$$

Any help is appreciated! I've already done the base case at zero. Thank you!

Answer 1

Hint: Inductive step:

Observe you have \begin{align} \sum^{n+1}_{r=0}\binom{n+1}{r}6^r =6^{n+1}+1+\sum^{n}_{r=1}\left\{\binom{n}{r-1}+\binom{n}{r}\right\}6^r \end{align}

Answer 2

If you have already proven the binomial formula

$$(a+b)^{n} = \sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^{r}$$

it is enough to set $a$ and $b$ to the right values. If you haven’t, you can follow Jacky Chong’s hint to prove it in this particular case.