Square of Triangular Numbers as Sum of Triangular Numbers
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Theorem
- ${T_n}^2 = T_n + T_{n - 1} T_{n + 1}$
where $T_n$ denotes the $n$th triangular number.
Proof
\(\ds T_n + T_{n - 1} T_{n + 1}\) | \(=\) | \(\ds \frac {n \paren {n + 1} } 2 + \paren {\frac {\paren {n - 1} n} 2 \frac {\paren {n + 1} \paren {n + 2} } 2}\) | Closed Form for Triangular Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 n^2 + 2 n + \paren {n^2 - n} \paren {n^2 + 3 n + 2} } 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 n^2 + 2 n + n^4 - n^3 + 3 n^3 - 3 n^2 + 2 n^2 - 2 n} 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n^4 + 2 n^3 + n^2} 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {n \paren {n + 1} } 2}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds {T_n}^2\) | Closed Form for Triangular Numbers |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $15$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $15$