Formula for b ... b ( b ( a ) + c ) + c) + ... c 1) Let say we have a number A.  
2) We multiply it by B and add C to it. 
3) We repeat this action for N times.
For example for N = 3, A = 1, B = 3, C = 1, We have:
3 ( 3 ( 3 ( 1 ) + 1 ) + 1 ) + 1 ) = 40
What kind of progression do we have? Is it arithmetic or geometric? Is there any formula for the nth element or sum of the sequence until the nth element?

We want to look at the recurrence,
$$x_{n+1}=bx_{n}+c \tag{1}$$
With some initial value $x_0=a$. 
We may notice that there is a number $L$ such that,
$$L=bL+c \tag{2}$$
In the case $b \neq 1$, this number is $\frac{c}{1-b}$. We shall assume that, otherwise the solution is easy.
Subtracting the second equation from the first gives,
$$(x_{n+1}-L)=b(x_{n}-L)$$
So,
$$x_{n}-L=b^{n}(x_{0}-L)$$
(If your having trouble seeing this define $a_n=x_n-L$ then $a_{n+1}=ba_{n}$ etc) 
Which tells us that,
$$x_{n}=L+b^{n}(a-L)$$
Or more explicitly,
$$x_{n}=\frac{c}{1-b}+b^{n}(a-\frac{c}{1-b})$$
In the case $b=1$ obviously (and simple to prove),
$$x_{n}=x_{0}+cn=a+cn$$
