I recall learning in school how to convert arithmetic and geometric sequence formulas between recursive and explicit, but I don't remember learning a systematic method to approach it. For example, let $a_n=2a_{n-1}+2$ where $a_0=5$. How would I convert this to an explicit form defined in terms of $n$? Beyond this specific example, is there a generalized method to do so?
[Math] How to write recursive formula as explicit (specifically, exponential)
exponential functionfunctionsrecursive algorithmssequences-and-series
Best Answer
There is a systematic way of solving something like $a_n=2a_{n-1}+2a_{n-2}$, which is called a linear recurrence relation. For example this pdf shows you how to do it.
But for your specific example you have a constant $2$, so it's not a linear recurrence. However you can rewrite it as $a_n+2=2(a_{n-1}+2),a_0+2=7$ where it's obvious that $a_n+2=7\cdot 2^n$ so $a_n=7\cdot 2^n-2$.
For a completely general recurrence there isn't really a systematic way to convert it to a formula, but you can always try tricks like the above (linear recurrence, substitution) to see if they work.