MAT511 homework,         Sept. 24, 2003

1. Use induction to prove that, for all natural numbers $n$,
   $$1 + 4 + 7 + \ldots + (3n-2) = \frac{n(3n-1)}{2}$$

2. Prove that for any natural number $n$, $n^3 + 5n + 6$ is divisible by 3.

3. If a set $A$ has $n$ elements, prove that its power set ${\mbox{${\mathcal{P}}\left(A\right)$}}$ has $2^n$ elements.

4. Let $P_1$, $P_2$, ...$P_n$ be $n$ points in a plane, no three of which are collinear. Prove (by induction) that the number of line segments joining all pairs of points is $(n^2-n)/2$.