Can't figure out mathematical induction for these problems.

1. 1.1! + 2.2! + 3.3! + + . ! = ( + 1)! 1 for all 1

2. 1 2 + 2 2 2 3 + 2 4 2 5 + + (1) 2 = 2 +1 (1) +1 3 for all 0

3. 3 > 2 + 3 for all 2 [15 pts.] 4. 1 + 3 + 9 + 27 + + 3 = 3 +11 2 for all 0

5. 1 + 4 + 7 + 10 + + (3 2) = (31) 2 for all 1 [15 pts.] 6. (2 + 1) = 3 21 2 = for all positive integers .