Question 3 Prove each of the following statements using mathematical induction.

Question

Question 3

Prove each of the following statements using mathematical induction. For each statement, answer the following questions. (4 pt.) Complete the basis step of the proof by showing that the base case is true. (4 pt.) What is the inductive hypothesis? (4 pt.) What do you need to show in the inductive step of the proof? (8 pt.) Complete the inductive step of the proof. a. b. c. d. 5. Prove that 32n - 1 is divisible by 4 for any positive integer n Recall that an integer y is divisible by an integer x with x integer i such that y-x 0 if and only if there exists an

Explanation / Answer

P(n) = 32n-1

basis step - it should be true for base case i.e n=1

(a) P(1) = 32*1 - 1 = 8 which is divisible by 4.

(b) In the in ductive hypothesis we assume the function works for a given number K that is P(K) is true.

(1) - P(K) = 32*k - 1 is true i.e 32*k - 1 is divisible by 4

(c) in the inductive step we need to show that if P(k) is true then it should be also true for P(K+1)

hence with the help of the base case and the (1) we will prove that P(K+1) is true

(d) P(K+1) = 32*(k+1) - 1

= 32*k * 32 - 1 = 9* 32*k - 1

= 8*32*k + 32*k - 1

in above both the terms 8*32*k is divisible by 4 since 8 is divisible by 4.and 32*k - 1 is divisible by 4 through the inductive hypothesis.hence the above entire term is divisible by 4.therefore 32*(k+1) - 1 is also divisible by 4.

since the base case and inductive hypothesis steps are verified we can say that 32*n -1 is divisible by 4 for any positive integer.

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