question no3

Exercises 1. There are infinitely many stations on a train route. Sup- pose that the train stops at the first station and suppose f) Explain why these steps show that this formula is true whenever n is a positive integer that it the train stops at a station, then it stops at the next 4. Let P(n) be the statement that 13 +23 +... + station. Show that the train stops at all stations. (n(n +1)/2)2 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of 2. Suppose that you know that a golfer plays the first hole of a golf course with an infinite number of holes and that if this golfer plays one hole, then the golfer goes on to play the next hole. Prove that this golfer plays every hole on the proof. c) What is the inductive hypothesis? d) What do you need to prove in the inductive step? e) Complete the inductive step, identifying where you the course. Use mathematical induction in Exercises 3-17 to prove sum mation formulae. Be sure to identify where you use the in- ductive hypothesis. use the inductive hypothesis 3. Let P(n) be the statement that 122 + f) Explain why these steps show that this formula is true whenever n is a positive integer. n(n + 1 ) (2n + )/6 for the positive integer n. a) What is the statement P(1)? 5. Prove that 12+32+5(2n +1)2 (n+1) (2n + 1(2n +3)/3 whenever n is a nonnegative integer 6. Prove that 1.11+2.2! + . . . + n , n! = (n + 1)!-1 b) Show that P(1) is true, completing the basis step of the proof. c) What is the inductive hypothesis? d) What do you need to prove in the inductive step? e) Complete the inductive step, identifying where you whenever n is a positive integer. whenever n is a nonnegative integer (-7)+ 1)/4 whenever n is a nonnegative integer 7. Prove that 3 +3.5+3.52+. +3.5-3(5+1-1)/4 8. Prove that 2-2 . 7 + 2-72-+ 2(-7)"-(1- use the inductive hypothesis