Editing Tetrahedral number (section)

===Proofs of formula===
This proof uses the fact that the {{mvar|n}}th triangular number is given by
:<math>T_n=\frac{n(n+1)}{2}.</math>
It proceeds by [[Mathematical induction|induction]].

;Base case
:<math>Te_1 = 1 = \frac{1\cdot 2\cdot 3}{6}.</math>

;Inductive step
:<math>\begin{align} 
Te_{n+1} \quad 
&= Te_n + T_{n+1} \\
&= \frac{n(n+1)(n+2)}{6} + \frac{(n+1)(n+2)}{2}  \\
&= (n+1)(n+2)\left(\frac{n}{6}+\frac{1}{2}\right) \\
&= \frac{(n+1)(n+2)(n+3)}{6}.
\end{align}</math>

The formula can also be proved by [[Gosper's algorithm]].