User:Leigh.Samphier/Common/Induction Example
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Proof
Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
- $\ds n^2 = \sum_{j \mathop = 1}^n \paren {2 j - 1}$
Basis for the Induction
$\map P 1$ is true, as this just says $1^2 = 1$.
This is our basis for the induction.
Induction Hypothesis
Now we need to show that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is our induction hypothesis:
- $\ds k^2 = \sum_{j \mathop = 1}^k \paren {2 j - 1}$
Then we need to show:
- $\ds \paren {k + 1}^2 = \sum_{j \mathop = 1}^{k + 1} \paren {2 j - 1}$
Induction Step
This is our induction step:
\(\ds \paren {k + 1}^2\) | \(=\) | \(\ds k^2 + 2 k + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 1}^k \paren {2 j - 1} + 2 k + 1\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 1}^k \paren {2 j - 1} + 2 \paren {k + 1} - 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 1}^{k + 1} \paren {2 j - 1}\) |
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\ds \forall n \in \N: n^2 = \sum_{j \mathop = 1}^n \paren {2 j - 1}$
$\blacksquare$