# Newton's binomial theorem with induction

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?

• I edited your formula, but I had to do some guessing because it wasn't clear what you meant. Please check if what I wrote is the formula you had in mind – Alessandro Codenotti Oct 4 '15 at 9:26
• Yes it was that formula. How do you write this formulas here? Is there a program? – WatsonHolmes Oct 4 '15 at 12:38
• see here for a good introduction – Alessandro Codenotti Oct 4 '15 at 13:10

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)