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Consider the following sequence: 0, 6, 9, 9, 15, 24, . . .. Let the first...

Consider the following sequence: 0, 6, 9, 9, 15, 24, . . .. Let the first term of the sequence, a1 = 0, and the second, a2 = 6, and the third a3 = 9. Once we have defined those, we can define the rest of the sequence recursively. Namely, the n-th term is the sum of the previous term in the sequence and the term in the sequence 3 before it: an = an−1 + an−3. Show using induction that every integer in this sequence is divisible by 3: for all n ≥ 1,an mod 3 = 0.

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