Sequences & Series Questions and Answers

ΔABC is right-angled at A and has AB < AC. Point D is on BC so that AB = BD. Point L is the midpoint of AD. Let P be the point on the circumcircle of ΔADC so that ∠APB = 90°. (a) Prove that B, P, L, and A are concyclic. (b) Prove that ∠LPC = 90°.

Let f be the triangle wave function defined by f(x): = π-|x| for -π < x ≤ π, and extended to be 2π-periodic on the real number line. Find the Fourier series of this function in complex exponential form. You should verify the Fourier coefficients of f are: f(n)= {π/2 if n = 0 0 if n≠0 and even 2/(πn²) if n is odd

Researchers have shown that the number of successive dry days that occur after a rainstorm for a particular region is a random variable that is distributed exponentially with a mean of 9 days Complete parts (a) and (b) a. What is the probability that 13 or more successive dry days occur after a rainstorm? b. What is the probability that fewer than 3 dry days occur after a rainstorm?

Let V be any vector space (potentially infinite dimensional). Prove that (V/W)* ~ W^o

Consider F and C below. F(x, y, z) = yze^xzi + e^xzj + xye^xZk, C: r(t) = (t² + 2)i + (t² − 1)j + (t² − 5t)k, 0 ≤ t ≤ 5 (a) Find a function fsuch that F = Vf. f(x, y, z) =

A bank has set aside a maximum of $25 million for commercial and home loans. Every million dollars in commercial loans requires 2 application forms while every million dollars in home loans requires 3 forms. The bank cannot process more than 72 forms at this time. The bank's policy is to loan at least four times as much for home loans as for commercial loans. At least $10 million will be used for these two types of loans. The bank earns 8% on commercial loans and 10% on home loans. What amount of money should be allotted for each type of loan to maximize the interest income?

Alice deposits $500 in her account at the rate 6% compounded annually. What will be the amount in her account after 8 years? Round off the answer to the nearest tens value.

jessica replaces letters in the calculation SW-EE+T with numbers 5,11,13,18,19 and then calculates the result. The same letters are replaced by the same numbers and different letters by different numbers. What is the smallest possible result that is greater than zero? A. 7 B. 2 C. 4 D. 9. E. 5. Please show working, what numbers can be used in the equation

The perimeter of a rectangle is 72 cm and the width is 13 cm. What is the length of the rectangle? Round to the nearest tenth if necessary.

Paloma pays $42 to join the ski team. She pays $12 each time she enters a competition. Describe the domain and range of this situation. Domain is all natural numbers greater than or equal to 30 Range is all natural numbers greater than or equal to 12 Domain is all natural numbers greater than or equal to 54 Range is all natural numbers Domain is all natural numbers Range is all natural numbers Domain is all natural numbers Range is all natural numbers greater than or equal to 42

What is the corresponding system of equations? X₁ + x2 + x3 S₁ = 75 x₁ + x2 + x3 S₂ = 48 X₁ + X2 S3 = 53 (Type + if the variable is to be added or if it must be subtracted.)

You are driving on the interstate, keeping up with the speed of traffic. Suddenly, you are hit with torrential rains. You cannot feel the road beneath your tires anymore. You cannot steer. Your brakes are not working as they normally do. What just happened?! You are hydroplaning. When a car hydroplanes, it drives on top of a layer of water. Since it has lost contact with the roadway, it cannot easily change direction or stop. Hydroplaning usually occurs at high speeds on roads with deep puddles. Engineers have determined a formula that predicts when hydroplaning will occur: v = 10.35√p, where v is the speed (in miles per hour) at which a car will hydroplane, and p is the tire pressure (in PSI). This means that cars with underinflated tires are more likely to hydroplane. Your task: answer the following questions and show all your work. 1. Mike is driving a Toyota Camry. He's inflated his tires to the dealer- recommended level of 32 PSI. At what speed will Mike's car start hydroplaning?

Test suppose that for five years at 6% toward the purchase of a car. you barrow $11,000 Use PMT= P( =) [L-(H+) to find the monthly payments and total interest for the loan. The monthly payment is $ cent-) (Do not round until final answer. Then round to nearest The total interest for the loan is ?

Find sin 2x given that cos x = Give exact solution. 5 13 and x is in quadrant IV.

Determine whether the function is one-to-one. If it is, find a formula for its inverse. 3 f(x)=x³-8 ... Is the function one-to-one? O Yes O No Select the correct choice below and fill in any answer boxes within your choice. OA. The inverse function is f(x) = 1 OB. There is no inverse function.

Quest A. How much money would you need to invest today at 4.5% annual interest compounded quarterly to have $10,000 in 30 months? (Accurate to the nearest dollar.} B. You have $20,000 today and want to earn $1,000 in interest over the next three years. What annual interest rate would you have to earn on your investment, assuming it is compounded monthly? (Express answer to nearest tenth of percent.) Formula: nt A (t) = P(1 + 2) t P=Principal; r = annual rate of interest; n= #times interest compounded annually, t= #years A = Value of investment after t years. (For each part show initial equation to be used to answer question. Then final answer.}

The exponential function N= 2000 x 1.75%, where d is measured in decades, gives the number of individuals in a certain population. (Round decimal answers to two decimal places unless otherwise indicated and report percents to the nearest percent.) (a) Calculate N(1.8). N(1.8) = Explain what your answer means. The population is about (b) What is the percentage growth rate per decade? 96 people after (c) What is the yearly growth factor rounded to three decimal places? What is the yearly percentage growth rate? % (d) What is the growth factor for a century? What is the percentage growth rate per century? % years.

Find all solutions of the equation 2 sin²z - cos x = 1 in the interval [0, 2m). The answer is 2₁ = 0 x₂ = with 1 <2< 23. and 23 =

Use the cofunction identities to find an angle 0 between 0 and 90° that makes the statement true. csc0 sec(40+50°) = 8= = (Type an integer or a simplified fraction.) BOX

A Room is 200' long and 150 wide and the 8' high celling is smooth and flat. A spot type heat detector has been selected that is listed for a 50° maximum spacing between detectors. According to NFPA 72 how many detectors will be needed to protect this room? 200/50x150/50 -

Solve for t. e^-0.13t = 0.47 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. ... A. t= B. The solution is not a real number.

Rewrite 6 sin(x) + 4 cos(x) as A sin(x + φ)

For any n, nj means the product of all the positive even integers before n For example, 18j=18*16*--*2 What is the greatest prime factor of 22j+20j?

Given the two points (1,12.5) and (2,15.625), find the equation of the exponential function that passes through these two points.

In the equation y =56/x, y varies inversely as x. When x = 7, y = 8. What is the value of y when x = 14? When x = 14, y = (Simplify your answer.)

Consider the sequence defined recursively by s₁ = 1 and sn= s n-1 + 2(n-1) for n ≥ 2. (a) Give the first four terms of the sequence, s1, s2, s3 and s4. s1: s2: s3: s4 (b) is the sequence monotone? Enter Yes or No. (c) Is the sequence bounded below? If so, enter a number M such that Msn for every n > 0. If not, enter No. (d) is the sequence bounded above? If so, enter a number M such that sn≤M for every n ≥ 0. If not, enter No. (e) Does the sequence converge? If so, enter the limit it converges to. If not, enter No.

Let f=((-4,2), (-1,4), (3,0)) and g=((-3,3), (1,3), (2.-7), (4,-1)). Find fog. fog= (Use a comma to separate ordered pairs as needed.).

The total cost of attending a university is $15,700 for the first year. A student's parents will pay one-fourth of this cost. An academic scholarship will pay $3,000. Which amount is closest to the minimum amount the student will need to save every month in order to pay off the remaining cost at the end of 12 months?

The perimeter of a rectangular outdoor patio is 66 ft. The length is 5 ft greater than the width. What are the dimensions of the patio?

What is the total cost to repay a $5500 loan with a 5% interest rate for a term of 2 years?

Convert the following degree measure to radians. 145°20'

Find the angle of least positive measure (not equal to the given measure) coterminal with A. A=28° 59'

Sketch an angle 8 in standard position such that 8 has the least possible positive measure and the point (-3,4) is on the terminal side of 0. Then find the exact values of the six trigonometric functions for 0. sin 8- (Simplify your answer. Type an integer or a fraction.) The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Find the present value that will grow to $2000 if the annual interest rate is 9.5% compounded quarterly for 6 yr.

If you don't have a calculator, you may want to approximate (2, 187.0156)4/7 by (2187)4/7 = 81. Use the Mean Value Theorem to estimate the error in making this approximation.

Consider the two groups and G = {< x, y > |xª¹ = 1; y² = 1; xy³ = yx}. G' =< x,y> x¹ = 1; y² = x²; x³y = Prove that G is not isomorphic to G'.

Let f(x)=x² + 2x and g(x)=9-x. Find (f/g)(x) and (f/g)(8).

Solve and graph the solution set. 1<t-4 and t-2≤8

State the definition of a p-series and the p-series theorem from class. Give an example of a p-series that converges and one that diverges.

Question 9. Let H: R 3⇒R 3 be the mapping defined by H((a, b, c)) = (a + 2b, a - c, 2a + 2b + 2c). (i) Find H((1,0,1)). Then determine whether H is a linear transformation.

A box has edges from (0, 0, 0) to (3, 1, 1), (1, 3, 1), and (1, 1, 3). Find its volume and also find the area of each parallelogram face.

Approximately what range of heights can be reasonably reached from a 24-foot ladder (i.e., 24 feet cannot reasonably be reached because the ladder will fall backward and 0 feet cannot reasonably be reached because the ladder is flat on the ground)? What assumptions have you made to arrive at your answer?

Use the information below to determine which equation belongs to which sibling, as well as the object which was thrown and the age order. 1. The youngest launched his object from atop the playset. 2. Ann climbed a tall tree to launch her object. 3. Damian's object went the highest. 4. Debbie's object was in the air for the shortest amount of time. 5. The arrow was shot by the oldest. 6. The golf ball went 87 feet into the air. 7. Ann shot her arrow higher than the tennis and bowling balls. 8. The bowling ball reached its maximum height in just under 1 second. 9. Damian's ball was in the air twice as long as the 2nd oldest sibling. 10. The object of the youngest brother was in the air for more than 4 seconds

Suppose Quinton borrows a student loan of $8,500 at an interest rate of 6.5%. Once graduating from college, he is expected to pay that money back over the course of 20 years. Blank #1: Find Quinton's monthly payment. Round to the nearest cent. Blank #2: How much money will Quinton pay in total? Round to the nearest cent.

A national park has a population of 3000 bears in the year 2016. Conservationists are concerned because the bear population is declining at a rate of 5% per year. If the population continues to decrease at this rate, how long will it take until the population is only 1500 bears?

Let x = (2, -3λ²), y = (λ, -1) and z= (λ -3,1) be vectors in R². Part(a) Find the value(s) of λ such that y and z are parallel. Justify your answer. Part(b) Find the value(s) of λ such that x and y are orthogonal. Justify your answer.

Let x = (-2, 3a²), y = (-a, 1) and z = (3-a,-1) be vectors in R². Part(a) Find the value(s) of a such that y and z are parallel. Part(b) Find the value(s) of a such that x and y are orthogonal.

dy/dx=2y (2x4 + 3 cos x - 2) a. Find the general solution. b. Find the particular solution using the boundary condition y(0) = 15.

The median starting salary for an entry-level position at a certain business is $56,000 per year, but the salary could differ from the median salary by as much as $5,000 based on the experience of the applicant. If the starting salary is x, which of the following inequalities describes all possible starting salaries for the entry- level position, and only those salaries? x-5,000 ≤ 56,000 x-56,000 ≤ 10,000 |x-5,000| ≤ 56,000 |x-56,000| ≤ 5,000

a. Suppose that a stock's price is rising at a rate of 8% per year and that it continues to increase at this rate. If the value of one share of this stock is $15 now, find the value of one share of this stock three years from now. b. The value of a car decreases after it is purchased. Suppose that the value of the car depreciates according to an exponential decay model. Suppose that the value of the car is $10.000 at the end of 7 years and that its value has been decreasing at the rate of 7% per year. Find the value of the car when it was new.