I need to prove by induction the following equality. I did the inductive hypothesis part, but I don't get it when $n=1$. Any help/ hints are greatly appreciated.
Be $x\neq 1$ and a real number, prove for all $n \in N$ $$\frac{x^n-1}{x-1} = (x^{n-1} + x^{n-2} +\cdots+ x^2 + x + 1)$$
This is what I've gotten so far:
Assuming this is true for $n$: $$\frac{x^n-1}{x-1} = (x^{n-1} + x^{n-2} +\cdots+ x^2 + x + 1)$$
Then prove it is true for $n+1$
$$\frac{x^{n +1}-1}{x-1} = (x^{n} + x^{n-1} +\cdots+ x^2 + x + 1)$$
Proof: $$\frac{x^n-1}{x-1} = (x^{n-1} + x^{n-2} +\cdots+ x^2 + x + 1) \Rightarrow$$ $$x^n-1 = (x-1) (x^{n-1} + x^{n-2} +\cdots+ x^2 + x + 1)$$
And I need to prove that:
$$\begin{align}x^{n+1}-1 &= (x-1)(x^{n} + x^{n-1} +\cdots+ x^2 + x + 1)\\ x^{n+1}-1 &= x\cdot x^n-1\\ &= x\cdot x^n -x^n+ x^n-1\\ &=(x-1)x^n+(x^n-1) \end{align}$$ Replacing $x^n-1$, $$=(x-1)x^n+(x-1)(x^{n-1} +\cdots+1)\\ =(x-1)(x^n+x^{n-1} +\cdots+1)$$
And that's proved. But what about $n=1$? Thanks!