Use strong induction to prove that that the following holds for all natural numbers n and all real numbers \(x \not = 1\).
\(\displaystyle \sum_{i=0}^n x^i =\frac{x^{n+1}-1}{x-1}\)
Stuck? See the hints.
When you have a proof that seems close to correct, look at the solution.
Suppose that we draw n lines in the plane, in general position (no lines are parallel, no point belongs to more than two lines). The lines divide up the plane into a set of regions. Prove the following claim, for any positive integer n:
Claim: we can color these regions with two colors, such that adjacent regions (i.e. touching along an edge) never have the same color.
Stuck? See the hints.
When you have a proof that seems close to correct, look at the solution.