# Arithmetic Sequence in Harmonic Sequence

If Harmonic Sequence

$$H_n=\left\{ {1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},...} \right\}$$

How can I find an arithmetic sequence using the above sequence?

Actually, I have found one by inspection. $$\left\{{\frac{1}{3},\frac{1}{4},\frac{1}{6}}\right\}\\ \text{It is an arithmetic sequence with common difference}=-\frac{1}{6}$$ However, I have no idea on finding the other arithmetic sequence with more term. Thank you.

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Note that your example can be written over the common denominator 12 as $$\frac{4}{12}, \frac{3}{12}, \frac{2}{12}.$$ This suggests starting with a (decreasing) arithmetic progression of natural numbers, then finding common denominator, and turning it into fractions.
EDIT: Try the sequence $$\frac{n-k}{n!}$$ for $k=0,1,2...;$ for example $n=5$ gives $$\frac{1}{24},\frac{1}{30},\frac{1}{40},\frac{1}{60},\frac{1}{120}.$$