Exercise 14.

Prove the following statements by mathematical induction:

1. \(n! \lt n^n\) for \(n \ge 2\)

2. \(\d\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} +\frac{1}{3\cdot 4}+\cdots + \frac{1}{n\cdot(n+1)} = \d\frac{n}{n+1}\) for all \(n \in \Z^+\text{.}\)

3. \(4^n - 1\) is a multiple of 3 for all \(n \in \N\text{.}\)

4. The greatest amount of postage you cannot make exactly using 4 and 9 cent stamps is 23 cents.

5. Every even number squared is divisible by 4.