Let D3 be the symmetry group of an equilateral triangle. Show that the subgroup H ⊂ D3 consisting of those symmetries which are rotations is a normal subgroup.

Prove that if a sequence is bounded, then limsup sn is a real number.

Find the particular solution:

y″−4y′+4y=(−11.5e^(2t))/(t^(2)+1)

Show that a point x is an accumulation point of a set E if and only if for every ε > 0 there are at least two points belonging to the set E ∩ (x − ε, x + ε).

A binary string is a “word” in which each “letter” can only be 0 or 1

Prove that there are 2^n different binary strings of length n.

Note:

• Your goal is to produce a properly constructed proof by induction, but this does not mean you have to follow
• Mathematical induction, step-by-step..
• Write the statement with n replaced by k
• Write the statement with n replaced by k+1.
• Identify the connection between the kth statement and the (k+1)th statement.
• Complete the induction step by assuming that the n=k version of the statement is true, and using this assumption to prove that the n=k+1 version of the statement is true.
• Complete the induction proof by proving the base case.
• point-by-point.
• ANOTHER IMPORTANT NOTE:
• Make sure you are addressing the given statement: "there are 2ⁿ different binary strings of length n” !!! So your solution should primarily be discussing binary strings. Yes, the fact that 2^k+1 = 2^k∙2 will be useful in your solution, but it should not be the sole content of your solution, as by itself that equality is a fact about exponents, not a fact about binary strings.

Helen purchased a house for $450,000. She made a downpayment of 15% of the value of the house and received a mortgage for the rest of the amount at 5.50% compounded semi-annually for 15 years. The interest rate was fixed for a 5-year term.

a. Calculate the size of the monthly payments.

Round to the nearest cent

b. Calculate the principal balance at the end of the 5-year term.

Round to the nearest cent

c. Calculate the size of the monthly payments if after the first 5-year term the mortgage was renewed for another 5-year term at 5.25% compounded semi-annually?

Round to the nearest cent

use bisection to find the real root x = sin(x) + 1 with the initial guesses of x l = 0 and x u = 3.

perform the computation until the approximate error falls below 5%

Let a be a positive element in an ordered field. Show that if n is an odd number, a has at most one nth root; if n is an even number, a has at most two nth roots.

Hampton Commission on the Arts (HCA) is planning on advertising its upcoming events. A local newspaper claims that the ads on the paper can reach 27,000 art lovers, and it costs the HCA $6,000. Advertising on a local radio $10,000, and it is estimated to reach an audience of 39,000 people. HCA also considers google ads, which costs $15,000 will supposedly reach 52,000 arts customers. The breakdown of the audience for each type of commercial is as follows (charts below):

HCA has determined some marketing guidelines in order to improve outreach efforts. It wants to reach at least 200,000 potential arts customers. Past experience shows that older people are more likely to buy tickets than younger people; the number of potential customers over 35 should be at least 50% more than the potential customers under 35. The council also believes women are more likely to initiate ticket purchases to the arts than men, so it wants its advertising audience to be at most 40% male.

1) Construct a mathematical model for HCA to determine the number of ads of each type it should use at the minimum cost.

2) Solve the model using Excel Solver

(a) Luis Mahla purchases a Porsche Boxster for $49,700 and finances the entire amount at an annual interest rate of 6.3% for 8 years. Find the monthly payment. Assume the sales tax is 6% of the purchase price and the license fee is 1% of the purchase price. (Round your answer to the nearest cent.)

(b) After becoming a commercial pilot, Lorna Kao decides to purchase a Cessna 182 for $64,975. Assuming the sales tax is 4.9% of the purchase price, find each of the following.

(1) What is the total cost, including sales tax?
(2) If Lorna makes a down payment of 18% of the total cost, find the down payment.
(3) Assuming Lorna finances the remaining cost at an annual interest rate of 7.15% for 10 years, find the monthly payment.

At the beginning of each football season, the coaching staff at Vista High School must vote to decide which players to select for the team. They use the weighted voting system {7: 6, 5, 1}. In this voting system, the head coach A has a weight of 6, the assistant coach B has a weight of 5, and the junior varsity coach C has a weight of 1. Compute the Banzhaf power index for each of the coaches. (Round your answers to the nearest hundredth.)

BPI(A) =

BPI(B) =

BPI(C) =

Let D(x, y) be the predicate defined on natural numbers x and y as follows: D(x, y) is true whenever y divides x, otherwise it is false. Additionally, D(x, 0) is false no matter what x is (since dividing by zero is a no-no!). Let P(x) be the predicate defined on natural numbers that is true if and only if x is a prime number. 1. Write P(x) as a predicate formula involving quantifiers, logical connectives, and the predicate D(x, y). Assume the domain to be natural numbers.

Hint 1: n is prime if and only if the only numbers that divide it are 1 and n.

Hint 2: You might have to use conditionals.

2. Consider the proposition “There are infinitely many prime numbers”. Express the proposition as a predicate formula using quantifiers, logical connectives and the predicate P(x). Assume the domain to be natural numbers. Note that you don’t need to use the answer from the previous part in this problem; you can write your answer in terms of P(x).

3. Write the negation of the predicate formula obtained in part 2. Make sure you take the negation all the way in so that it sits right next to P(x) in the final expression.

Only want to know what the answer for 3 should be

in 2013 the estimated world population 7.1 billion use the doubling of 70 years to predict the population 2027, 20153 and 2103?

Suppose A^2= A , where A is an n by n matrix. Use Jordan canonical form to thaw that A is diagonalizable.

B = {red,red,green,purple}
C = {red,{green},red,{red,green},purple,{green,green,red,purple}}

A.) What is the power set of C?

B.) Is B ∈ P(C)?