XoaX.net's MathML: Induction Proof

We want to prove that $P (n) = 1 + 2 + 3 + \dots + (n - 1) + n = \frac{n (n + 1)}{2}$ , for all $n \in ℕ$ .

We start by confirming the equation for the case $n = 1$ .
For $P (1)$ , we have $P (1) = 1 = \frac{1 \times 2}{2}$

Then assume the formula is true for the $k$ th case:
That is, $P (k) = 1 + 2 + 3 + \dots + (k - 1) + k = \frac{k (k + 1)}{2}$

For the $(k + 1)$ th case, we have
$\begin{matrix} P (k + 1) & = & 1 + 2 + 3 + \dots + (k - 1) + k + (k + 1) \\ = & P (k) + (k + 1) \\ = & \frac{k (k + 1)}{2} + (k + 1) \\ = & \frac{k (k + 1)}{2} + \frac{2 (k + 1)}{2} \\ = & \frac{(k + 1) (k + 2)}{2} \end{matrix}$