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Arithmetic – Harmonic Progression

Consider the harmonic sequence

\displaystyle 1, \frac{1}{2}, \frac{1}{3} ,  \cdots, \frac{1}{n} , \cdots

Prove that if we pick dinstinct terms of the above sequence we can construct an arithmetic progression sequence of as large (finite) length as we want.

Solution

We just observe that

\displaystyle \frac{1}{n!}, \frac{2}{n!}, \cdots, \frac{n!}{n!}

are all distinct terms of the sequence.

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