I apologize for the confusion. Here's a simpler explanation with the steps in LaTeX.

The $n$th triangle number is the sum of the first $n$ positive integers. It can be written as:

$$T(n) = 1 + 2 + 3 + ... + n$$

To prove that the formula for $T(n)$ is $\frac{n(n+1)}{2}$, we can use the following steps:

Step 1: Write the sum forwards and backwards and add them together:

T(n) &= 1 + 2 + 3 + ... + n \\
T(n) &= n + (n-1) + (n-2) + ... + 1 \\
\hline \\
2T(n) &= (n+1) + (n+1) + (n+1) + ... + (n+1)
\end{aligned}$$
There are $n$ terms in the sum, so we can simplify this to:
$$2T(n) = n(n+1)$$
Step 2: Divide both sides by 2 to get the formula for $T(n)$:
$$T(n) = \frac{n(n+1)}{2}$$
Here's the same explanation in a LaTeX block, which you can copy and paste into a LaTeX editor (such as Overleaf or ShareLaTeX) to visualize the equations:
```
The nth triangle number is the sum of the first n positive integers. It can be written as:
$$T(n) = 1 + 2 + 3 + ... + n$$
To prove that the formula for T(n) is n(n+1)/2, we can use the following steps:
Step 1: Write the sum forwards and backwards and add them together:
$$\begin{aligned}
T(n) &= 1 + 2 + 3 + ... + n \\
T(n) &= n + (n-1) + (n-2) + ... + 1 \\
\hline \\
2T(n) &= (n+1) + (n+1) + (n+1) + ... + (n+1)
\end{aligned}$$
There are n terms in the sum, so we can simplify this to:
$$2T(n) = n(n+1)$$
Step 2: Divide both sides by 2 to get the formula for T(n):
$$T(n) = \frac{n(n+1)}{2}$$
```
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