Sum of Adjacent Sequences of Triangular Numbers

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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.


Sources

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