Prove the following statements by using the definition of convergence for sequences of
real numbers.

a) If {cn} is a sequence of real numbers and {cn} converges to 1 then {1/(cn+1)} converges to 1/2

b) If {an} and {bn} are sequences of real numbers and {an} converges A and {bn} converges to B and B is not equal to 0 then {an/bn} converges to A/B

*NUMBER THEORY*

1.Find all the possible solutions for the following diphantine equations by using the euclidian algorithim. You must show all the process to get credit.

a.           3x + 5y = 7

b.           3x − 12y = 7

c.           1990x − 173y = 11

d.           21x + 48y = 6

e.           2x + 3y + 5z = 11

Winkler Furniture manufactures two different types of china cabinets: a French provincial model and a Danish Modern model. Each cabinet produced must go through three departments: carpentry, painting, and finishing. The table below contains all relevant information concerning production times per cabinet produced and production capacities for each operation per day, along with net revenue per unit produced. The firm has a contract with an Indiana distributor to produce a minimum of 300 of each cabinet per week (or 60 cabinets per day). Owner Bob Winkler would like to determine a product mix to maximize his daily revenue.

Formulate as an LP problem and obtain the revenue      

what is the principal on a $10,000 loan at a rate of 7% number of compounding per year is 12 term in years is 18 months what is the payment and the total pay of interest paid
how do you find the monthly payment.

1. A consumer electronics company is planning to introduce a new device. After careful consideration of costs, (e.g., there is a fixed cost of $5 million for developing the item),  the projected state of the economy, etc., the marketing manager came up with the following payoff table (in $millions)

1. Calculate the regret table below. SHOW ALL WORK TO GET FULL CREDIT. (6 pts.)

2. Calculate the minimax regret below. SHOW ALL WORK TO GET FULL CREDIT. (3 pts.)

1. Using the following table of relation between X & Y (Y is the independent Variable):

xi -2 2 4 5 4 6 7 8 11

yi -4 1 3 1 3 8 10 20 30

a. Find the estimated regression equation. Show ALL your calculations and the appropriate formula. WRITE Formula in all the questions below and substitute for the numbers, demonstrate your work.

b. Calculate R2. Comment on the Goodness of the Fit. Is it a good fit? Why?

c. Test the hypothesis regarding the significance of the slope β1. (Use α = 0.05) Show ALL the calculations necessary to get your answer. Hint: You need to calculate t by using all the formulas needed for it using t in the Formula Sheet.

d. Calculate the values of F, using formula for F statistics. e. Draw the scatter diagram and also your estimated regression line in a diagram.

1. A survey of 500 high school students was taken to determine their favorite chocolate candy. Of the 500 students surveyed, 64 like Snickers, 125 like Twix, 141 like Reese's Peanut Butter Cups, 30 like Snickers and Twix, 61 like Twix and Reese's Peanut Butter Cups, 43 like Snickers and Reese's Peanut Butter Cups, and 18 like all three kinds of chocolate candy. How many students like Twix and Reese's Peanut Butter Cups only?

2. A survey of 391 children given at a local elementary school showed that 200 like chocolate ice cream, 175 like pistachio ice cream, and 141 do not like chocolate or pistachio ice cream. How many children like at most one kind of ice cream mentioned in the survey?

Given the function f(x,y)=2x^3 + 6xy^2 - 3y^2 - 150x. Find the local minimums, maximums and saddle points of f(x,y).

(a) What is the maximum degree of a vertex in a simple graph with n vertices?

(b) What is the maximum number of edges in a simple graph of n vertices?

(c) Given a natural number n, does there exist a simple graph with n vertices and the maximum number of edges?

Diagonalize the matrix (That is, find a diagonal matrix D and an invertible matrix P such that

A=PDP−1.

(Do not find the inverse of P). Describe all eigenspaces of A and state the geometric and algebraic multiplicity of each eigenvalue.

A=

Supply: logP= -15.00+1.50 logQ

Demand: logP=5020-0.75 logQ

1. Create a table as shown. The first column contains values of Q from 500 to 10,000, in intervals of 500

2. Plot the supply and demand on the diagram. Label right X and Y axes

New supply: logP=-15.00+1.50 logQ+0.90

1. Create 2 new columns of‘New supply (log P)’ and 'new supply (P) for the new supply

2. Add the new supply curve to the existing diagram. Label the new supply curve

Show that if G is a group of order np where p is prime and 1 < n < p, then G is not simple.

(Please do not use Sylow theorem)

Belden, Inc. acquires 30 percent of the outstanding voting shares of Sheffield, Inc. on January 1, 2017, for $306,000, which gives Belden the ability to significantly influence Sheffield. Sheffield has a net book value of $784,000 at January 1, 2017. Sheffield's asset and liability accounts showed carrying amounts considered equal to fair values except for a copyright whose value accounted for Belden's excess cost over book value in its 30 percent purchase. The copyright had a remaining life of 16 years at January 1, 2017. No goodwill resulted from Belden's share purchase. Sheffield reported net income of $182,000 in 2017 and $244,000 of net income during 2018. Dividends of $64,000 and $90,000 are declared and paid in 2017 and 2018, respectively. Belden uses the equity method. On its 2018 comparative income statements, how much income would Belden report for 2017 and 2018 in connection with the company's investment in Sheffield? If Belden sells its entire investment in Sheffield on January 1, 2019, for $412,000 cash, what is the impact on Belden's income? Assume that Belden sells inventory to Sheffield during 2017 and 2018 as follows. What amount of equity income should Belden recognize for the year 2018? Year Cost to Belden Price to Sheffield Year-End Balance (at Transfer Price) 2017 $36,000 $60,000 $20,000 (sold in following year) 2018 34,160 61,000 38,000 (sold in following year)