Sum of Sequence of Products of Consecutive Integers/Proof 2

Proof
Observe that:

Thus:
 * $3 i \left({i + 1}\right) = b \left({i + 1}\right) - b \left({i}\right)$

where:
 * $b \left({i}\right) = i \left({i + 1}\right) \left({i - 1}\right)$

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

Hence the result.