Question

For m = 1, 2, …, 4m+2 is a multiple of ________

(a) 3

(b) 5

(c) 6

(d) 2

This question was addressed to me by my college director while I was bunking the class.

Question is from Principle of Mathematical Induction topic in section Induction and Recursion of Discrete Mathematics

Answer 1

Correct option is (d) 2

Explanation: For n = 1, 4 * 1 + 2 = 6, which is a multiple of 2. Assume that 4m+2 is true for m=k and so 4k+2 is true based on the assumption. Now, to prove that 4k+2 is also a multiple of 2 ⇒ 4(k+1)+2 ⇒ 2 * 4k – 4k + 6 ⇒ 2*4k+4 – 4k+2 ⇒ 2(4k+2) – 2(2k+1). Here, the first term 2(4k+2) is true as per assumption and the second term 2(4k+2) is must to be a multiple of 2. Hence, 4(k+1)+2 is a multiple of 2. So, by induction hypothesis, (4m+2) is a multiple of 2, for m = 1,2,3,…