## Try It Now

Work these exercises to see how well you understand this material.

### Solutions

- Answer:

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

Induction: Assume:*B*(*k*) = 3*k*+ 2 for some*k*≥ 0.*B*(*k*+ 1) =*B*(*k*) + 3

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

= (3*k*+ 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.