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.
Best Answer
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.
This likely would give long progressions.
EDIT: Try the sequence $$\frac{n-k}{n!}$$ for $k=0,1,2...;$ for example $n=5$ gives $$\frac{1}{24},\frac{1}{30},\frac{1}{40},\frac{1}{60},\frac{1}{120}.$$