What is the general formula for a sequence that is neither arithmetic nor geometric? If a sequence is such that the next term is given by $a_n*a +b$, what is the general formula of the given sequence?
e.g.: $-$ for $a=4,\;b=6$, the sequence is:
$1,\quad1*4+6=10,\quad10*4+6=46,\quad46*4 +6=190,\;\dots$
What would the formula for the general term be?
(I'm not necessarily asking specifically for this example but rather for a general sequence with the next term defined by $a_n*a +b$).

You have recursion $$x_{n+1} = a x_n + b \qquad a \neq 1, \quad b\neq 0$$
Note that $$ x_{n+1} - \frac{b}{1-a} = a x_n + b - \frac{b}{1-a} = a x_n - \frac{ab}{1-a} = a\big(x_{n+1} - \frac{b}{1-a}\big)$$
which means that $y_n = x_n - \frac{b}{1-a} $ is a geometric series:
$$ y_{n+1} = a y_n$$
It can be solved $$ y_n = a^n y_0$$
and you get $$ x_n = y_n + \frac{b}{1-a} = a^n(x_0 - \frac{b}{1-a}) + \frac{b}{1-a}$$
$$ x_n = a^n x_0 + b\frac{1-a^n}{1-a}$$
