MathDB

Observe that \frac{1}{1}= \frac{1}{2}+\frac{1}{2}; \frac{1}{2}=\frac{1}{3}+\frac{1}{6}; \frac{1}{3}=\frac{1}{4}+\frac{1}{12}; \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)}.

[hide="Remark."] It seems that this is a two-part problem.

Problem Statement