Write the first five terms of the following sequence and obtain the corresponding series:
Question:

Write the first five terms of the following sequence and obtain the corresponding series:

$a_{1}=3, a_{n}=3 a_{n-1}+2$ for all $n>1$

Solution:

$a_{1}=3, a_{n}=3 a_{n-1}+2$ for all $n>1$

$\Rightarrow a_{2}=3 a_{1}+2=3(3)+2=11$

$a_{3}=3 a_{2}+2=3(11)+2=35$

$a_{4}=3 a_{3}+2=3(35)+2=107$

$a_{5}=3 a_{4}+2=3(107)+2=323$

Hence, the first five terms of the sequence are 3, 11, 35, 107, and 323.

The corresponding series is 3 + 11 + 35 + 107 + 323 + …