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We consider the set E of all fractions

\frac{1}{n}

, where

n

is a natural number. A maximal arithmetic progression of length

k

of the set E is an arithmetic progression of

k

terms such that all its terms belong to the set E, and it is impossible to extend it to the right or to the left with another element of E.
For example,

\frac{1}{20}, \frac{1}{8}, \frac{1}{5}

, is an arithmetic progression in E of length

3

, and it is maximal, since to extend it towards to the right you have to add

\frac{11}{40}

, which does not belong to E, and to extend it to the left you have to add

\frac{-1}{40}

which does not belong to E either.
Prove that for every integer k> 2, there exists a maximal arithmetic progression of length

k

of the set E.

6

Problems(1)