Please include your work.

1. (7 points) Prove using induction that whenever n is a positive integer. 2. (8 points) Prove using mathematical induction that the following inequality holds for n 2 5: 3. (5 points) This is a variant of the Josephus problem. Josephus was one of the leaders of a Jewish revolt against Rome in the year 66. Assume that n prisoners were sitting in a circle and numbered clockwise from 1,2, ..., n. Proceeding clockwise, every other prisoner was killed. Prisoner 2 was killed first, then prisoner 4, and this continued until there was only one prisoner left. This lone remaining prisoner was not killed and considered the lone survivor. The goal was to stand in the correct position initially such that you would be the lone survivor. Denote the position of the lone survivor as J(n) where n is the number of people sitting in the circle. As an example: J(5) 3 (Order of deaths: 2, 4,1,5) Prove using induction that J(2)- for ali2