# 2. Find the sum:

$$
\frac{2}{1 \cdot 2 \cdot 3}+\frac{2}{2 \cdot 3 \cdot 4}+\frac{2}{3 \cdot 4 \cdot 5}+\ldots+\frac{2}{2008 \cdot 2009 \cdot 2010}
$$
Solution:

Notice that $\frac{2}{n \cdot(n+1) \cdot(n+2)}=\frac{1}{n}-\frac{1}{n+1}-\frac{1}{n+1}+\frac{1}{n+2}$.

From this, it follows that the required sum is:

$\frac{1}{1}-\frac{1}{2}-\frac{1}{2}+\frac{1}{3}+\frac{1}{2}-\frac{1}{3}-\frac{1}{3}+\frac{1}{4}+\frac{1}{3}-\frac{1}{4}-\frac{1}{4}+\frac{1}{5}+\ldots+\frac{1}{2008}-\frac{1}{2009}-\frac{1}{2009}+\frac{1}{2010}$

It is easy to see that all terms, except the first two and the last two, cancel out, thus we get:

$\frac{1}{1}-\frac{1}{2}-\frac{1}{2009}+\frac{1}{2010}=\frac{1009522}{2019045}$

Answer: $1009522 / 2019045$.