## anonymous 4 years ago What is the next term in the sequence below? 1/2, 2/6, 3/18, 4/54, ____

1. anonymous

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

2. anonymous

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

3. anonymous

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

4. mathslover

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

5. mathslover

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

6. anonymous

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

7. mathslover

but it do have common difference...

8. anonymous

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