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Solutions

  1. Answer:

    Basis: B(0) = 3 · 0 + 2 = 2, as defined.

    Induction: Assume: B(k) = 3k + 2 for some k ≥ 0.

    B(k + 1) = B(k) + 3

    = (3k + 2) + 3 by the induction hypothesis

    = (3k + 3) + 2

    = 3(k + 1) + 2 as desired

  2. Answer: Imagine drawing line k in one of the infinite regions that it passes through. That infinite region is divided into two infinite regions by line k. As line k is drawn through every one of the k − 1 previous lines, you enter another region that line k divides. Therefore k regions are divided and the number of regions is increased by k. From this observation, we get P(5) = 16.

  3. Answer: For n greater than zero, M(n) = M(n − 1) + 1, and M(0) = 0.