Question

Use induction to prove that n(n + 1)(n + 2) is a multiple of 3 for all integers n ≥ 1.

Answer 1

To prove that n(n + 1)(n + 2) is a multiple of 3 for all integers n ≥ 1, we use induction by following the base case and inductive step.

Step-by-step explanation:

To prove that n(n + 1)(n + 2) is a multiple of 3 for all integers n ≥ 1 using induction:

1. Base Case: Show that the statement is true for n = 1:
   n(n + 1)(n + 2) = 1(1 + 1)(1 + 2) = 1(2)(3) = 6 which is a multiple of 3.
2. Inductive Step: Assume the statement is true for n = k.
   So, k(k + 1)(k + 2) is a multiple of 3.
3. Show that the statement is true for n = k + 1:
   (k + 1)(k + 1 + 1)(k + 1 + 2) = (k + 1)(k + 2)(k + 3)
4. Since k(k + 1)(k + 2) is a multiple of 3 (as assumed in step 2), it can be expressed as 3m for some integer m.
5. Substituting 3m for k(k + 1)(k + 2) in (k + 1)(k + 2)(k + 3), we get:
   (k + 1)(k + 2)(k + 3) = 3m(k + 1)(k + 3) which is a multiple of 3.

Since the statement is true for n = 1 and assuming it is true for n = k implies it is true for n = k + 1, we can conclude that n(n + 1)(n + 2) is a multiple of 3 for all integers n ≥ 1 by induction.