Let $$S(n+1)=S(n)+cb^{n+1}, n=0,1, 2 ..., c\in\mathbb{R}, b\neq 1$$
Prove by induction:
$$S(n)=a+c\left(\frac{b^{n+1}-1}{b-1}\right), n=0,1,2... $$
Let $p(n):S(n)=a+c\left(\frac{b^{n+1}-1}{b-1}\right) $
First Step
In this case I dont' now if I should prove $p(0)$ or $p(1)$ Anyway I am not sure how do this step because I don't know $S(0)$ or $S(1)$. Can you help me please?
Edited
Is valid this? $$S(1)=S(0)+cb$$ $$p(1)=a+c\left(\frac{b^{2}-1}{b-1}\right)=a+c(b+1)=a+c+cb$$
If $S(0)=a+c$ then $p(1) $ is true.
Second Step
Assume: $p(n):S(n)=a+c\left(\frac{b^{n+1}-1}{b-1}\right) $ (Induction Hypothesis)
Show:$p(n+1):S(n+1)=a+c\left(\frac{b^{n+2}-1}{b-1}\right)$
By definition we know that $$ p(n+1):S(n+1)=S(n)+cb^{n+1}$$
Using the Induction Hypthosesis: $$ p(n+1):S(n+1)=a+c\left(\frac{b^{n+1}-1}{b-1}\right)+cb^{n+1}=a+c\left(\frac{b^{n+1}-1}{b-1}+b^{n+1}\right)$$ $$p(n+1)=a+c\left(\frac{b^{n+2}-1}{b-1}\right)$$
Q.E.D
Now without induction
$$ s_{n+1} = s_n + c b^{n+1}\\ s_{n} = s_{n-1} + c b^{n}\\ \cdots \\ s_0 = a $$
then
$$ s_{n+1} = a+c\sum_{k=0}^{n}b^{k} = a+c\left(\frac{b^{n+1}-1}{b-1}\right) $$
Inductively
First step
$$ S_0 = a+c = a+c\left(\frac{b-1}{b-1}\right) $$
Second step
Assuming
$$ S_n = a + c\left(\frac{b^{n+1}-1}{b-1}\right) $$
then
$$ S_{n+1} = S_n + c b^{n+1} = a + c\left(\frac{b^{n+1}-1}{b-1}\right)+c b^{n+1} = a+c\left(\frac{b^{n+2}-1}{b-1}\right) $$
Q.E.D