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windsylph
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.
the sequence is as follows: 14-2=12 86-14=12*6 518-86=12*6*6 and so on......
for ones like this I usually work my way backwards \[a_n = 6a_{n-1} + 2=6(6a_{n-2}+2)+2=6^2a_{n-2}+6\cdot 2+2\] \[=6^2(6a_{n-3}+2)+6\cdot 2+2=6^3a_{n-3}+6^2\cdot 2+6\cdot 2+2=\cdots\]
\[a_{n} = 6^{n-1}a_{n-(n-1)}+6^{n-2}.2+...+6.2+2\] \[a_{n} = 6^{n-1}.2 + 6^{n-2}.2 + ... + 6.2 + 6^{0}.2\] \[a_{n} = 2\frac{6^{n}-1}{6-1}\]