Difference of Two Powers/Proof 1

Proof
Let $\displaystyle S_n = \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$.

This can also be written:
 * $\displaystyle S_n = \sum_{j \mathop = 0}^{n - 1} b^j a^{n - j - 1}$

Consider:
 * $\displaystyle a S_n = \sum_{j \mathop = 0}^{n - 1} a^{n - j} b^j$

Taking the first term (where $j = 0$) out of the summation, we get:
 * $\displaystyle a S_n = \sum_{j \mathop = 0}^{n - 1} a^{n - j} b^j = a^n + \sum_{j \mathop = 1}^{n - 1} a^{n - j} b^j$

Similarly, consider:
 * $\displaystyle b S_n = \sum_{j \mathop = 0}^{n - 1} a^j b^{n - j}$

Taking the first term (where $j = 0$) out of the summation:
 * $\displaystyle b S_n = \sum_{j \mathop = 0}^{n - 1} a^j b^{n - j} = b^n + \sum_{j \mathop = 1}^{n - 1} a^{n - j} b^j$

This is equal to:
 * $\displaystyle b^n + \sum_{j \mathop = 1}^{n - 1} a^j b^{n - j}$

by Permutation of Indices of Summation.

So: