Sum of Sequence of Products of Consecutive Integers/Proof 2

Proof
Observe that:

Thus:
 * $3 i \paren {i + 1} = b \paren {i + 1} - b \paren i$

where:
 * $b \paren i = i \paren {i + 1} \paren {i - 1}$

This can therefore be used as the basis of a telescoping series, as follows:

Hence the result.