Equivalence of Definitions of Second Pentagonal Number

Theorem
The following definitions of a second pentagonal number are equivalent:

Proof
Let $P' \left({n}\right)$ denote the $n$th second pentagonal number.

We have:

But:
 * $P' \left({n}\right) = \dfrac {n \left({3 n + 1}\right)} 2$

is the definition of a second pentagonal Number by definition 2.

Hence the result.