Question

What is the next term in the sequence below?

1/2, 2/6, 3/18, 4/54, ____

Answer 1

\[\frac{ 1 }{ 2 }, \frac{ 2 }{ 6 },\frac{ 3 }{ 18 },\frac{ 4 }{ 54 },\frac{ ? }{ ? }\]

Answer 2

1/2 + 1/3 + 1/6 + 1/9??

Answer 3

We note:
\[
\frac{a_n}{b_n}=\frac{a_{n-1}+1}{3b_{n-1}}
\]So:
\[
\frac{5}{162}
\]Is our next term.

Answer 4

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

Answer 5

\[\large{\frac{4}{54}+\frac{-1}{6}= \frac{4-9}{54}=\frac{-5}{54}}\]

Answer 6

It's not an arithmetic series @mathslover , since:
\[
\lim_{n\to\infty}\left(\frac{a_n}{b_n}\right)=0
\]

Answer 7

but it do have common difference...

Answer 8

For the first three terms, check:
\[
\frac{4}{54}-\frac{3}{18}\ne -\frac{1}{6}
\]