Use mathematical induction to prove the statement is true for all positive integers n.The integer n3 + 2n is divisible by 3 for every positive integer n.
What will be an ideal response?
| Step 1: | Determine if S1, S2, and S3 are true. |
S2: 3 is a factor of 23 + 2*2 = 12
S3: 3 is a factor of 33 + 2*3 = 33
Step 2: Assume Sk to be true, then
Sk: 3 is a factor of k3 + 2k.
k3 + 2k = 3m
Step 3: Find Sk+1: 3 is a factor of (k + 1)3 + 2(k + 1).
Step 4: (k + 1)3 + 2(k + 1) = (k3 + 3k2 + 3k + 1) + (2k + 2)
(k + 1)3 + 2(k + 1) = k3 + 2k + 3k2 + 3k + 3
(k + 1)3 + 2(k + 1) = (k3 + 2k) + (3k2 + 3k + 3)
(k + 1)3 + 2(k + 1) = (3m) + (3k2 + 3k + 3)
(k + 1)3 + 2(k + 1) = 3(m + k2 + k + 1)
3 is a factor of 3(m + k2 + k + 1)
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