Problem:
Solution:
The right hand side sounds like something we can telescope, let's see:
$ \begin{eqnarray*} & & (a - b)\sum\limits_{k = 0}^{n-1}{a^{k}b^{n-1-k}} \\ &=& a\sum\limits_{k = 0}^{n-1}{a^{k}b^{n-1-k}} - b\sum\limits_{k = 0}^{n-1}{a^{k}b^{n-1-k}} \\ &=& \sum\limits_{k = 0}^{n-1}{a^{k+1}b^{n-1-k}} - \sum\limits_{k = 0}^{n-1}{a^{k}b^{n-k}} \\ &=& (a^n + \sum\limits_{k = 0}^{n-2}{a^{k+1}b^{n-1-k}}) - (\sum\limits_{k = 1}^{n-1}{a^{k}b^{n-k}} + b^n ) \\ &=& (a^n + \sum\limits_{k = 1}^{n-1}{a^{k}b^{n-k}}) - (\sum\limits_{k = 1}^{n-1}{a^{k}b^{n-k}} + b^n ) \\ &=& a^n - b^n \end{eqnarray*} $
The right hand side sounds like something we can telescope, let's see:
$ \begin{eqnarray*} & & (a - b)\sum\limits_{k = 0}^{n-1}{a^{k}b^{n-1-k}} \\ &=& a\sum\limits_{k = 0}^{n-1}{a^{k}b^{n-1-k}} - b\sum\limits_{k = 0}^{n-1}{a^{k}b^{n-1-k}} \\ &=& \sum\limits_{k = 0}^{n-1}{a^{k+1}b^{n-1-k}} - \sum\limits_{k = 0}^{n-1}{a^{k}b^{n-k}} \\ &=& (a^n + \sum\limits_{k = 0}^{n-2}{a^{k+1}b^{n-1-k}}) - (\sum\limits_{k = 1}^{n-1}{a^{k}b^{n-k}} + b^n ) \\ &=& (a^n + \sum\limits_{k = 1}^{n-1}{a^{k}b^{n-k}}) - (\sum\limits_{k = 1}^{n-1}{a^{k}b^{n-k}} + b^n ) \\ &=& a^n - b^n \end{eqnarray*} $
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