**8**

A statement S_{n} about the positive integers is given. Write statements S_{1}, S_{2}, and S_{3}, and show that each of these statements is true.

S_{n}: 1^{2} + 4^{2} + 7^{2} + . . . + (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 S_{1}, S_{2}, and S_{3}, and show that each of these statements is true.

S_{n}: 2 + 5 + 8 + . . . + ( 3*n *– 1) = *n*(1 + 3*n*)/2

**13**

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

2 is a factor of n^{2} – n + 2

**14**

A statement S_{n} about the positive integers is given. Write statements S_{1}, S_{2}, and S_{3}, and show that each of these statements is true.

S_{n}: 2 is a factor of n^{2} + 7n