Try It Now
Work these exercises to see how well you understand this material.
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
- 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.
- Answer: For n greater than zero, M(n) = M(n − 1) + 1, and M(0) = 0.