How do I prove $1+x+x^2+\cdots+x^{n-1}=\frac{1-x^n}{1-x}$ using the induction method?

Question

I am having difficulties in solving the expression $1+x+x^2+\cdots+x^{n-1} =\frac{1-x^n}{1-x}$, $n\in \mathbb{N}$, using the induction method. How is the inductive step taken in this case?

Answer 1

Assume that the formula holds for some $n\in\mathbb N$. Then

$$\underbrace{1+x+\ldots + x^{n-1}}_{\text{formula holds for this part}}+x^n = \frac{1-x^n}{1-x} + x^n.$$

Can you finish now?

Answer 2

Assume that $1+x + \dots + x^{n-1} = \frac{1-x^n}{1-x}$.

Then

$$1+x+ \dots + x^{n} = (1+ x + \dots + x^{n-1}) + x^n = \frac{1-x^n}{1-x} + x^n$$ $$= \frac{1-x^n}{1-x} + \frac{x^n(1-x)}{1-x} = \frac{1-x^n + x^n - x^{n+1}}{1-x}= \frac{1-x^{n+1}}{1-x}$$