Mathematical Induction Exercises

Answer each problem. Show your answers to the GTA to get credit for today's recitation.

1. Prove by mathematical induction that $2^n > n^2$ for all integers $n \ge 5$.

2. Prove by mathematical induction $$\begin{align*} \frac{1}{1\cdot 3} + \frac{1}{3\cdot 5} + \cdots + \frac{1}{(2n-1)(2n+1)} = \frac{n}{2n+1} \end{align*}$$

3. Prove by mathematical induction that the following program computes $n\cdot m$.

public int mult(int n, int m) {
  if (n == 1)
    return m;
  return m + mult(n-1, m);
}

4. Prove by mathematical induction that $$1^2 + 2^2 + \cdots + n^2 = \frac{1}{6}n(n+1)(2n+1)$$

5. What is the Big-O notation for the growth rate of the execution time for this algorithm?

for (int i = 0; i < n; i++)
  for (int j = 0; j < i; j++)
    for (int k = 0; k < i; k++)
      x = x * 42;