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?

  • $\begingroup$ 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 $\endgroup$ – Alessandro Codenotti Oct 4 '15 at 9:26
  • $\begingroup$ Yes it was that formula. How do you write this formulas here? Is there a program? $\endgroup$ – WatsonHolmes Oct 4 '15 at 12:38
  • $\begingroup$ see here for a good introduction $\endgroup$ – Alessandro Codenotti Oct 4 '15 at 13:10

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)

  • $\begingroup$ Oh ok! I understood, I'm going to try again, thanks! $\endgroup$ – WatsonHolmes Oct 4 '15 at 12:39

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