Kizspy | Question: 21 (Choose 1 answer)
Find the correct order of the steps of the proof by induction of the following problem:
Problem: Prove that for all n ≥ 12 we have n=4a+5b with a, b non-negative integers.
Step 1. Suppose for some k ≥ 16 we have n=4x+5y with x, y non-negative integers for all n ≤ k.
Step 2. n=12 we have 12=4\times3+5\times0 n=13 we have 13=4\times2+5\times1 n=14 we have 14=4\times1+5\times2 n=15 we have 15 = 4x0 + 5x3
Step 3. So for all n ≥ 12 we have n=4a+5t with a, b non-negative integers.
Step 4. For n=k+1 we have k+1=1\times4+(k-3). Since 12\le k-3<k, by induction hypothesis we have k - 3 = 4a+5b for some non-negative integers a . Then k+1=4(a+1)+5b.
A. 2, 1, 4, 3
B. 1, 2, 4, 3
C. 2, 1, 3, 4
D. 1, 2, 3, 4
