0

I'm trying to prove using induction that $a^n +b^n = (a-b) \cdot \sum\limits_{k=1}^{n}a^{n-k}b^k-1$

So I have developed the expression for $n+1$ but I get to $a \cdot b^n - b^{n+1} + a^n - b^n$

And from here I can't simplify until $a^{n+1} + b^{n+1} \ldots$

Can someone help me?

Thanks

barak manos
  • 43,109
WatsonHolmes
  • 149

1 Answers1

0

Induction step (assuming a and b are non-zero, of course);

$ (a-b) \cdot \sum\limits_{k=1}^{n+1}a^{n+1-k}b^{k-1} =$

$ \sum\limits_{k=1}^{n+1}a^{n+2-k}b^{k-1} - \sum\limits_{k=1}^{n+1}a^{n+1-k}b^k=$

$ a^{2}\sum\limits_{k=1}^{n+1}a^{n-k}b^{k-1} - ab \sum\limits_{k=1}^{n+1}a^{n-k}b^{k-1}=$

$ (a^2 - ab)\sum\limits_{k=1}^{n+1}a^{n-k}b^{k-1}=$

$ a(a - b)\sum\limits_{k=1}^{n}a^{n-k}b^{k-1} + (a^2 - ab)a^{-1}b^n=$

$ a(a^n - b^n) + (ab^n - b^{n+1})=$

$ a^{n+1} - ab^n + ab^n - b^{n+1}=$

$a^{n+1} - b^{n+1}$

fleablood
  • 124,253