# Sum of Adjacent Sequences of Triangular Numbers

## Theorem

 $\ds T_1 + T_2 + T_3$ $=$ $\ds T_4$ $\ds T_5 + T_6 + T_7 + T_8$ $=$ $\ds T_9 + T_{10}$ $\ds T_{11} + T_{12} + T_{13} + T_{14} + T_{15}$ $=$ $\ds T_{16} + T_{17} + T_{18}$

and so on.

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

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

## Proof

 $\ds \sum_{k \mathop = n^2 + n - 1}^{n^2 + 2 n} T_n$ $=$ $\ds \sum_{k \mathop = 1}^{n^2 + 2 n} T_n - \sum_{k \mathop = 1}^{n^2 + n - 2} T_n$ $\ds$ $=$ $\ds H_{n^2 + 2 n} - H_{n^2 + n - 2}$ Definition of Tetrahedral Number $\ds$ $=$ $\ds \frac {\paren {n^2 + 2 n} \paren {n^2 + 2 n + 1} \paren {n^2 + 2 n + 2} } 6 - \frac {\paren {n^2 + n - 2} \paren {n^2 + n - 1} \paren {n^2 + n} } 6$ Closed Form for Tetrahedral Numbers $\ds$ $=$ $\ds \frac {n \paren {n + 1} \paren {n + 2} } 6 \paren {\paren {n + 1} \paren {n^2 + 2 n + 2} - \paren {n - 1} \paren {n^2 + n - 1} }$ $\ds$ $=$ $\ds \frac {n \paren {n + 1} \paren {n + 2} } 6 \paren {3 n^2 + 6 n + 1}$ $\ds$ $=$ $\ds \frac {n \paren {n + 1} \paren {n + 2} } 6 \paren {\paren {n + 3} \paren {n^2 + 3 n + 1} - \paren {n + 1} \paren {n^2 + 2 n + 2} }$ $\ds$ $=$ $\ds \frac {\paren {n^2 + 3 n} \paren {n^2 + 3 n + 1} \paren {n^2 + 3 n + 2} } 6 - \frac {\paren {n^2 + 2 n} \paren {n^2 + 2 n + 1} \paren {n^2 + 2 n + 2} } 6$ $\ds$ $=$ $\ds H_{n^2 + 3 n} - H_{n^2 + 2 n}$ Closed Form for Tetrahedral Numbers $\ds$ $=$ $\ds \sum_{k \mathop = 1}^{n^2 + 3 n} T_n - \sum_{k \mathop = 1}^{n^2 + 2 n} T_n$ Definition of Tetrahedral Number $\ds$ $=$ $\ds \sum_{k \mathop = n^2 + 2 n + 1}^{n^2 + 3 n} T_n$

$\blacksquare$

## Historical Note

David Wells states in Curious and Interesting Numbers ($1986$) that this result was pointed out by M.N. Khatri, but fails to give details.