A triangle number is a number that is the sum of all the integers from 1 up to a given integer n. The equation for finding the nth triangle number is given by the formula:

\begin{equation}
T_n = \frac{n(n+1)}{2}
\end{equation}

To prove this formula, we can use mathematical induction:

Basis step: For n = 1, the equation gives us T1 = 1 * (1 + 1) / 2 = 1, which is the first triangle number.

Inductive step: Assume that the formula holds for n = k, i.e., T_k = k * (k + 1) / 2.

Now we need to prove that the formula also holds for n = k + 1, i.e., T_{k + 1} = (k + 1) * (k + 2) / 2.

\begin{align}
T_{k + 1} &= T_k + (k + 1) \
&= \frac{k(k + 1)}{2} + (k + 1) \
&= \frac{k^2 + k}{2} + \frac{2k + 2}{2} \
&= \frac{k^2 + 3k + 2}{2} \
&= \frac{(k + 1)(k + 2)}{2} \
&= T_{k + 1}
\end{align}

Therefore, the formula holds for n = k + 1 as well.

Conclusion: By mathematical induction, we have shown that the formula T_n = n * (n + 1) / 2 holds for all positive integers n.
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