Question

Find a solution to the sequence 2, 14, 86, 518, 3110, ... . In other words, find a closed form of the recursive function a_n = 6a_{n-1} + 2.

Answer 1

the sequence is as follows:
14-2=12
86-14=12*6
518-86=12*6*6
and so on......

Answer 2

for ones like this I usually work my way backwards

\[a_n = 6a_{n-1} + 2=6(6a_{n-2}+2)+2=6^2a_{n-2}+6\cdot 2+2\]

\[=6^2(6a_{n-3}+2)+6\cdot 2+2=6^3a_{n-3}+6^2\cdot 2+6\cdot 2+2=\cdots\]

Answer 3

\[a_{n} = 6^{n-1}a_{n-(n-1)}+6^{n-2}.2+...+6.2+2\] \[a_{n} = 6^{n-1}.2 + 6^{n-2}.2 + ... + 6.2 + 6^{0}.2\] \[a_{n} = 2\frac{6^{n}-1}{6-1}\]