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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. $$ \left\{{\frac{1}{3},\frac{1}{4},\frac{1}{6}}\right\}\\ It\; is\; a\; Arithmetic\; Sequence\; with\; common\; difference=\;\frac{-1}{6} $$ However,i have no idea on finding the other arithemetic 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. This likely would give long progressions. EDIT: Try the sequence $$\frac{n-k}{n!}$$ for $k=0,1,2...;$ for example $n=5$ gives $$\frac{1}{24},\frac{1}{30},\frac{1}{40},\frac{1}{60},\frac{1}{120}.$$ |
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