How can I calculate summated math questions without using some kind of computer script or program? For example, how can I find the sum of $\frac 1 {(2 + n) \cdot (3 + n)}$ + ... for $n = 0$ to $n = 97$
 i.e.
$$\frac{1}{2 \cdot 3} + \frac 1 {3 \cdot 4} + ... + \frac 1 {99 + 100}$$

Note that $$\frac{1}{n(n + 1)} = \frac{1}{n} - \frac{1}{n + 1}$$
Hence, your second series can be rewritten as
$$\frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + ...$$
terminating, of course, at an appropriate point.

Similarly,
$$\frac{1}{(2 + n)(3 + n)} = \frac{1}{2 + n} - \frac{1}{3 + n}$$
