Hexagonal Number is Triangular Number

Theorem

Let $H_n$ be the $n$th hexagonal number.

Then $H_n$ is the $2 n - 1$th triangular number.

Proof

\(\ds H_n\)

\(=\)

\(\ds n \paren {2 n - 1}\)

Closed Form for Hexagonal Numbers

\(\ds \)

\(=\)

\(\ds \frac {2 n \paren {2 n - 1} } 2\)

\(\ds \)

\(=\)

\(\ds \frac {m \paren {m + 1} } 2\)

for $m = 2 n - 1$

Hence the result.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $45$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $45$