Sure, here's the equation and proof in LaTeX:

The equation for finding the nth triangle number is:

$$T_n = 1+2+3+\dots+n = \frac{n(n+1)}{2}$$

Proof:

Consider a pyramid of $n$ rows, where the $i$th row has $i$ objects. The total number of objects in the pyramid is:

$$T_n = 1+2+3+\dots+n$$

We can also think of the pyramid as consisting of two pyramids of height $(n-1)/2$, if $n$ is odd, or $n/2-1$, if $n$ is even.

For odd $n$, the first pyramid has $(n-1)/2$ rows, and the $i$th row has $i$ objects. The second pyramid has the same number of rows, but the $i$th row has $n-i+1$ objects. The total number of objects in both pyramids is:

$$2\left(1+2+3+\dots+\frac{n-1}{2}\right)+\frac{n+1}{2}$$

$$= \frac{n-1}{2}\cdot\frac{n+1}{2}+\frac{n+1}{2}$$

$$= \frac{(n-1)(n+1)}{2}+\frac{n+1}{2}$$

$$= \frac{n(n+1)}{2}$$

For even $n$, we can split the pyramid into two pyramids of height $n/2-1$, each of which has the same number of objects. Each of these pyramids can be considered as a pyramid of height $n/2$, where the $i$th row has $i$ objects. The total number of objects in both pyramids is:

$$2\left(1+2+3+\dots+\frac{n}{2}-1\right)$$

$$= n(n/2-1)$$

$$= \frac{n(n-2)}{2}$$

Adding the top row with $n$ objects, we get:

$$T_n = \frac{n(n-2)}{2}+n = \frac{n(n+1)}{2}$$

Thus, we have shown that the equation $T_n=\frac{n(n+1)}{2}$ holds for all $n\in\mathbb{N}$.

You can run this code in an online LaTeX processor like Overleaf (https://www.overleaf.com/) or sharelatex (https://www.sharelatex.com/).
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