Show that two sums are equal

Question

Show that $\sum_{k=0}^4\left(1+x\right)^k$ = $\sum_{k=1}^5 \left(5 \choose k\right)x^{k-1}$

I assume that this has something to do with the binomial theorem and a proof of that. But I can't take the first steps...

Answer 1

The LHS is a finite Geometric Series with the first term$=1,$ common ratio $=(1+x)$ and the number of terms $=5$

So, the sum is $$1\cdot\frac{(1+x)^5-1}{1+x-1}$$

Please expand using Binomial Expansion and cancel out $x$ to find it to be same as the RHS

Answer 2

Hint

Two polynomials with degree $4$ are equal if they have the same values in $5$ points.