I have an algorithm that generates terms as follows: $\begin{align*} \text{Term 1}: &2\times 3\times\left(\frac{1}{2}+\frac{1}{3}\right) + 4\\ \text{Term 2}: &2\times 3\times 4\times\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\right) + 3\times 5\\ \text{Term 3}: &2\times 3\times 4\times 5\times\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\right) + 3\times4\times 6\\ \text{Term 4}: &2\times 3\times 4\times 5\times 6\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}\right) + 3\times4\times 5\times 7 \end{align*}$ and so on.
The pattern in the first half is obvious; the pattern in the second half is a little trickier. It "splits" the last multiplier into (multiplier-1)*(multiplier+1) so 4 changes into 3*5 and then the 5 on that one changes into 4*6 and so on for the next term.
I am trying to find a clever way to say "sum terms 1-N" given this pattern