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Mathematical Induction

A method for proving mathematical hypotheses.

1.
Show hypothesis is true for some initial conditions
2.
Assume hypothesis is true for all values $\leq$ n
3.
Show hypothesis is true for larger value (usually, n + 1)



Example:

Prove \( \sum_{k=0}^n x^k = \frac{x^{n+1} - 1}{x-1} \).

Proof by mathematical induction:

1. For n = 0, x0 = 1, \( \frac{x-1}{x-1} = 1 \)

2. Assume \( \sum_{k=0}^n x^k = \frac{x^{n+1} - 1}{x-1} \)

3. Show \( \sum_{k=0}^{n+1} x^k = \frac{x^{n+2} - 1}{x-1} \)


\begin{eqnarray*}\sum_{k=0}^{n+1} x^k & = & \sum_{k=0}^n x^k + x^{n+1} \\
& = ...
...- 1 + x^{n+2} - x^{n+1}}{x-1} \\
& = & \frac{x^{n+2} - 1}{x-1}
\end{eqnarray*}




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