Finding the Rule For Simple Recurrence Relations

Suppose we want to find the rule or nth term for the sequence
3,11,35,107, 323,...
The first differences are
8, 22, 72, 216...
The second differences are
14, 50, 144
Neither the first or second differences are constant so the sequence is not linear or quadratic. It could however, be a simple recurrence relationship,  
\[u_{n+1} = a u_n +b\]
If it is we can easily found out. Divide each term by the previous on. In this case we obtain
\[\frac{11}{3}=6.666..., \frac{35}{11}=3.18181818..., \frac{107}{35}=3.0571428...,\frac{323}{107}=3.02...\]

Suceesive ratios are getting closer to 3. If the rule is of the forma above, we must have  
. To find b, substitute the first two erms into re relationship, giving
\[11=3 \times 3 +b \rightarrow b=2\]

The rule is then  
\[u_{n+1} = 3 u_n +2\]
  and it is easy to see that this generates the sequence.

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