8

A statement S_n about the positive integers is given. Write statements S₁, S₂, and S₃, and show that each of these statements is true.

S_n:   1² + 4² + 7² + . . . + (3n – 2)² =

S₁=0(6(0)^2-3(0)-1)/2=(3(1)-2)^2=1

S₂=1(6(1)^2-3(1)-1)/2=4^2=16

S₃=2(6(2)^2-3(2)-1)/2=7^2=49

9

A statement S_n about the positive integers is given. Write statements S_k and S_k+1, simplifying S_k+1 completely.

S_n: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3

10

Joely’s Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?

11

Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month and you need to make at least $3850 profit on them. How many of what type of printer should you order if you want to minimize your cost?

12

A statement S_n about the positive integers is given. Write statements S₁, S₂, and S₃, and show that each of these statements is true.

S_n:    2 + 5 + 8 + . . . + ( 3n – 1) = n(1 + 3n)/2

13

Use mathematical induction to prove that the statement is true for every positive integer n.

2 is a factor of n² – n + 2

14

A statement S_n about the positive integers is given. Write statements S₁, S₂, and S₃, and show that each of these statements is true.

S_n:   2 is a factor of n² + 7n