Sum of Adjacent Sequences of Triangular Numbers

Theorem
and so on.

The $n$th line of the pattern can be written as:


 * $\ds \sum_{k \mathop = n^2 + n - 1}^{n^2 + 2 n} T_n = \sum_{k \mathop = n^2 + 2 n + 1}^{n^2 + 3 n} T_n$