Prove by mathematical induction that $1 + 1/4 +\ldots + 1/4^n \to 4/3$ Please help. I haven't found any text on how to prove by induction this sort of problem: 
$$
 \lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} =  \frac{4}{3}
$$
I can't quite get how one can prove such. I can prove basic divisible "inductions" but not this. Thanks.

The below image can easily be made into an inductive proof.

To be precise, the above image first shows that
$$\dfrac12 + \dfrac14 + \dfrac18 + \cdots = 1$$ and then shows that
$$\dfrac1{2^2} + \dfrac1{(2^2)^2} + \dfrac1{(2^3)^2} + \dfrac1{(2^4)^2} + \cdots = \dfrac13$$
which implies
$$\dfrac1{4} + \dfrac1{4^2} + \dfrac1{4^3} + \dfrac1{4^4} + \cdots = \dfrac13$$
