Some Summations

1. Guess and prove a formula for the sum of the first n positive integers.
2. Guess and prove a formula for the sum of the first n positive odd integers.
3. Guess and prove a formula for the sum of the squares of the first n positive integers.
4. Guess and prove a formula for the sum of the cubes of the first n positive integers.
5. Guess and prove a formula for the sum of the fourth powers of the first n positive integers.