Question 3
Observe that
$$
\frac{1}{1} = \frac{1}{2} + \frac{1}{2} ; \quad \frac{1}{2} = \frac{1}{3} + \frac{1}{6} ; \quad \frac{1}{3} = \frac{1}{4} + \frac{1}{12} ; \quad \frac{1}{4} = \frac{1}{5} + \frac{1}{20}
$$
State a general law suggested by these examples, and prove it.
Prove that for any integer $n$ greater than 1 there exist positive integers $i$ and $j$ such that
$$
\frac{1}{n} = \frac{1}{i(i+1)} + \frac{1}{(i+1)(i+2)} + \frac{1}{(i+2)(i+3)} + \cdots + \frac{1}{j(j+1)}
$$
$$
\frac{1}{1} = \frac{1}{2} + \frac{1}{2} ; \quad \frac{1}{2} = \frac{1}{3} + \frac{1}{6} ; \quad \frac{1}{3} = \frac{1}{4} + \frac{1}{12} ; \quad \frac{1}{4} = \frac{1}{5} + \frac{1}{20}
$$
State a general law suggested by these examples, and prove it.
Prove that for any integer $n$ greater than 1 there exist positive integers $i$ and $j$ such that
$$
\frac{1}{n} = \frac{1}{i(i+1)} + \frac{1}{(i+1)(i+2)} + \frac{1}{(i+2)(i+3)} + \cdots + \frac{1}{j(j+1)}
$$