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. Show your work.

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

A statement S_n about the positive integers is given. Write statements S_kand S_k+1, simplifying S_k+1 completely. Show your work.

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

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?

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?

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. Show your work.

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

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

2 is a factor of n² – n + 2

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. Show your work.

S_n:   2 is a factor of n² + 7n

(i.) f(x)

(ii.) f(x)

(iii.) What can you conclude about f(x)? How is this shown by the graph?

(iv.) What aspect of costs of renting a car causes the graph to jump vertically by the same amount at its discontinuities?

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

8 + 16 + 24 + . . . + 8n = 4n(n + 1)

The following piecewise function gives the tax owed, T(x), by a single taxpayer on a taxable income of x dollars.

T(x) =

(i) Determine whether T is continuous at  6061.

(ii) Determine whether T is continuous at 32,473.

(iii) If T had discontinuities, use one of these discontinuities to describe a situation where it might be advantageous to earn less money in taxable income.

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 + 4 + 7 + . . . + (3n – 2) = n(3n – 1)/2

An artist is creating a mosaic that cannot be larger than the space allotted which is 4 feet tall and 6 feet wide. The mosaic must be at least 3 feet tall and 5 feet wide. The tiles in the mosaic have words written on them and the artist wants the words to all be horizontal in the final mosaic. The word tiles come in two sizes: The smaller tiles are 4 inches tall and 4 inches wide, while the large tiles are 6 inches tall and 12 inches wide. If the small tiles cost $3.50 each and the larger tiles cost $4.50 each, how many of each should be used to minimize the cost? What is the minimum cost?

The Fiedler family has up to $130,000 to invest. They decide that they want to have at least $40,000 invested in stable bonds yielding 5.5% and that no more than $60,000 should be invested in more volatile bonds yielding 11%. How much should they invest in each type of bond to maximize income if the amount in the stable bond should not exceed the amount in the more volatile bond? What is the maximum income?