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YorbelHunter
What is the next term in the sequence below? 1/2, 2/6, 3/18, 4/54, ____
\[\frac{ 1 }{ 2 }, \frac{ 2 }{ 6 },\frac{ 3 }{ 18 },\frac{ 4 }{ 54 },\frac{ ? }{ ? }\]
1/2 + 1/3 + 1/6 + 1/9??
We note: \[ \frac{a_n}{b_n}=\frac{a_{n-1}+1}{3b_{n-1}} \]So: \[ \frac{5}{162} \]Is our next term.
Arithmetic Progression : try to check whether there is common difference or not ... \[\large{\frac{2}{6}-\frac{1}{2} = \frac{2-3}{6}=\frac{-1}{6}=d}\] \[\large{\frac{3}{18}-\frac{2}{6}=\frac{3-6}{18}=\frac{-3}{18}=\frac{-1}{6}=d}\] hence the above sequence is of arithmetic progression
\[\large{\frac{4}{54}+\frac{-1}{6}= \frac{4-9}{54}=\frac{-5}{54}}\]
It's not an arithmetic series @mathslover , since: \[ \lim_{n\to\infty}\left(\frac{a_n}{b_n}\right)=0 \]
but it do have common difference...
For the first three terms, check: \[ \frac{4}{54}-\frac{3}{18}\ne -\frac{1}{6} \]