Exercises

Fun Class Activities:

1. Prove by induction that for all \( n \geq 1 \), the following formula is true:
   $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}.$$
2. Use induction to show that for all \( n \geq 1 \), we have \( 2^n \geq n + 1 \).
3. Prove by induction that \( 3^n > n^2 \) for all integers \( n \geq 5 \).
4. Show by induction that for all \( n \geq 1 \), the product
   $$\prod_{k=1}^n \left(1 + \frac{1}{k}\right) = n + 1.$$