Newton's binomial theorem with induction

Question

I'm trying to prove Newton's Binomial Theorem using induction.
I saw a resolution to help me doing it but in every resolution I found there's a point when they write

$\sum\limits_{k=0}^n \binom{n}{k}a^kb^{n-k+1}=b^{n+1}+\sum\limits_{k=1}^n\binom{n}{k}a^kb^{n-k+1}$

I don't understand some hidden steps between this 2 expressions... Can someone explain me?

Answer 1

There is no hidden step but just a shifting of index in summation notation.

If you imagine breaking up the left hand side into two pieces(k=0 and k=1 to n).

The first piece should be $\begin{pmatrix} n\\0\\ \end{pmatrix}$$a^0$ $b^{n-0+1}$ (this is just by substitute k=0) which equals to $b^{n+1}$, that's the first term of the R.H.S and the remaining part is just the second piece (k=1 to n)