Sequences & Series Questions and Answers

If θ =-1π/6, then sin (θ) =? cos(θ) =? Give exact values. No decimals allowed! Example: Enter sqrt(2)/2 for √2/2. With functions like sqrt, be sure to use function notation (parentheses). sqrt(2)/2 will work, but sqrt2/2 will not.

For f(x) = a*sin(bx), use: Amplitude = |a|; Period T = 2.π/|b| Section Width a.k.a. Distance between Critical Points = T/4 Without graphing the function y= -3sin(6x), determine its amplitude, period, and the distance between its critical points. Leave answers in exact form; type pi for π. amplitude = period = Distance between critical points =

14. Determine whether the series ∞ Σ (-1)ⁿ/{n^[(n+1)/n]} n=1 is conditional convergent, absolute convergent, or divergent.

Find an equation for the line perpendicular to 2x + 4y = 24 and goes through the point (3, -3). Write your answer in the form y = mx + b. y = ________

Suppose that there are weather patterns in a city. If it is sunny, there is a 20% chance that it will be rainy the next day. If it is raining, there is a 40% chance that it will be sunny the next day. (1) Diagonalize the stochastic matrix. (J) Find and simplify Mr. (K) If it is sunny today, what is the probability that it will be sunny after n days?

Let ABC be triangle and I its incenter. Denote D the midpoint of BC and E a the midpoint of arc BC containing vertex A. Prove angle IDB = angle IEA.

Without graphing the function y = 10 cos (5x), determine its amplitude and period. Leave answers in exact form; type pi for . amplitude = period =

The position(m)-time(sec) data for a moving body is given: 0 0.25 0.5 0.75 1.25 1.5 13 10.2 17.2 26.2 33.1 39.1 Approximate the velocity at time t = 1.25 sec by using central differences (b) Approximate the acceleration at time t = 1.25 sec by using central differences

Without graphing the function y = 6 sin(10x), determine its amplitude and period. Leave answers in exact form; type pi for π. amplitude = period=

You want to purchase an automobile for $25,865. The dealer offers you 0% financing for 60 months or a $3,260 rebate. You can obtain 6% financing for 60 months at the local bank. Which option should you choose? Rebate 0% financing How much money will you save per month? (Round to two decimal places.)

Solve the system of linear equations by graphing. y = 2x 6x + y = 8

2a. The doubling time for dorkism is 4 weeks. That is, with a given amount of dorkism in your system, it will double every 4 weeks, so you'll be twice as dorky as you were 4 weeks ago. Let D(t) be the function name and the initial value be D, to show a model for this in weeks. 2b. How much dorkiness would you have at the end of a sixteen week semester?

Find the first four terms of the sequence. f(n)=5n-2 a₁ =

Find the next 3 terms of the arithmetic sequence 5, 15, 25, 35, The next 3 terms are

Write 4 additional terms of the geometric sequence 5, 10, 20, ... . The next 4 terms in the geometric sequence are

Write the first four terms of the sequence. an = an-1 +4, a₁ = 6 What is the first term? a₁ =

Find the first four terms of the sequence with first term 9 and nth term an = 4an-1 +5. a₁ = (Simplify your answer.)

Find the sum of the first 10 terms of the geometric sequence 2, 4, 8... (Type an integer or a decimal.)

4. Suppose the a function of two variables f(x, y) satisfies f = af₁ + bfy Prove that if there is a constant M >0 such that f(x, y)| < M for all (x, y), then f is a constant.

Match each equation below to the curve it represents. Each answer should be A, B, C, D, E, F, or G. CURVES A. Circle, B. Ellipse, C. Point, D. Parabola, E. Empty Set, F. Intersecting lines, G. Hyperbola, EQUATIONS a) 4x²-4y² + 8x + 12y-5=0 b) 4x²-3xy-18 = 0 c) x² + xy + y2²-6=0 d) 9x² +4y²+72x-16y + 160=0 e) 9x² +4y² +72x-16y+124=0 f) 3x² + 3y² - 6x +12y +60=0 g) y²-5x-4y-6=0 h) x² +y²-2x + 2y + 1 = 0

Write an equation for a line perpendicular to y = -5x - 5 and passing through the point (-10, 0). y = ________

Find the distance, d, between (-1,2) and (3,4), and find the midpoint, M, between these points. a) d=√20, M=(1,3) b) d=√52, M=(1,3) c) d=8, M=(3,2) d) d=1, M=(1,1)

Find the rules for the composite functions fo g and go f. f(x)=√(x+2); g(x) =1/(x-2) fog=_______ gof=_______

Find an equation for the line parallel to y = -x + 9 and goes through the point (4,- 9). Write your answer in the form y = mx + b. y = __________

For the following differential equation, determine (without solving) an interval on which the solution of the equation is certain to exist and be unique. Justify your answer. (Int)y' + y = eᵗ cost, y(1/2)= 2

Factor out the greatest common factor in the expression. 3x⁴ - 33x³ + 15x² Select the correct choice below and fill in any answer boxes within your choice. A. 3x⁴-33x³ + 15x² = _________ B. The polynomial is prime.

Find an equation of the line passing through the pair of points. Write the equation in the form Ax+By = C. (9,2), (8,-6) Choose the equation of the line in the form Ax+By = C. A. 8x+y=70 B. - 8x+y=-70 C. -x+y=70 D. x-8y=70

Factor out the greatest common factor in the expression. 3x⁴-21x³ +33x² Select the correct choice below and fill in any answer boxes within your choice. A). 3x⁴-21x³ +33x² B). The polynomial is prime.____

A job placement agency advertised that last year its clients, on average, had a starting salary of $29,500. Assuming that average refers to the mean, which of the following claims must be true based on this information? Note: More than one statement could be true. If none of the statements are true, mark the appropriate box. a)Last year at least one of their clients had a starting salary of exactly $29,500. b)Two years ago some of their clients had a starting salary of at least $29,500. c)Last year some of their clients had a starting salary of less than $34,000. d)Last year more than half of their clients had a starting salary of at least $29,500. e)Last year all of their clients had a starting salary of $29,500 or above. f)None of the above statements are true.

Factor the difference of two squares. x² - 121 Select the correct choice below and fill in any answer boxes within your choice. A. x²-121=_________ B. The polynomial is prime.

(a) Find the rational zeros and then the other zeros of the polynomial function f(x) = 2x³ + 7x² + 98x + 343, that is, solve f(x) = 0. (b) Factor f(x) into linear factors. (a) Select the correct choice below and fill in any answer box(es) within your choice. (A) There are only rational zeros and they are ____. (B) There is only one rational zero, __, and the other zeros are __. (C) There are no rational zeros. The other zeros are ____. (b) The factorization of f(x) into linear factors is ____.

A 26-ft ladder leans against a building so that the angle between the ground and the ladder is 80°. How high does the ladder reach on the building? Round to the hundredths. (2 decimal places)

Write-2-log₄ +log₄(x + 1) + log, y as a single logarithm with coefficient 1. log₁(x + 1)^1/(1/2)y/16x

Simplify. Assume that all variables represent positive real numbers. -√180x⁸y¹⁹

Mary invested her $45,000 bonus in an account earning 7.1% annual interest that is compounded continuously. How long will it take the account to triple in value? Round your answer to the nearest tenth.

Divide as indicated. y²-64/y÷y+8/y-8 Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) A) y²-64/y÷y+8/y-8=__, y≠ ___. B) y²-64/y÷y+8/y-8=___,no numbers must be excluded.

Graph the linear equation. x-3y = -9 Use the graphing tool to graph the equation.

Factor the given polynomial. x² + 5x+6 Select the correct choice below and, if necessary, fill in the answer box within your choice. A. x² + 5x + 6 =_______ B. The polynomial is prime.

Given the definition of the main window of the periodic function g(t) = t-t², -1 < t < 2, g(t + 3) = g(t) The window to the right of this window has equation g(right) = q₇t² +q₈t+ q₉, ** <t<q₁₀

Can a triangle have sides with the given lengths? Explain. 5.) 4 m, 8 m, 6 m. 6.) 2.6 ft, 4.1 ft, 6.7 ft

The product of two consecutive room numbers is 110. Find the room numbers. The number of the first room, or the smaller room number, is ____ The number of the second room is ____

If tant =-5/12 and 3π/2< t < 2 π, find sint, cost, sect, csct, cot t. Enter the exact answers. sin t = cos t = sec t = csc t= cot t =

Three sequences are defined as follows: a₀= 1 and an+1 = bn+cn b₀ = 2 and bn+1 = an + Cn, c₀= 3 and Cn+1 = an + bn. (a) Use a spreadsheet to find the values of an, bn and cn as far as n = 10. (b) Find a general formula for the sum an + bn + Cn. (c) Determine the least value of n for which an +bn+Cn exceeds 100 000.

Add or subtract. 5√48 - 2√32 + √27 5√48 - 2√32 + √27 = ____

Replace the radical with a fractional exponent. Do not simplify. (x⁷)¹/⁵ = ________

Chris can be paid in one of two ways. Plan A is a salary of $360 per month, plus a commission of 9% of gross sales. Plan B is a salary of $708 per month, plus a commission of 3% of gross sales. For what amount of gross sales should Chris select plan A? Lets represent the amount of gross sales in dollars. Select the correct choice below and fill in the answer box to complete your choice. (Simplify your answer. Type an integer or a decimal.) A. The solution set is (8182 B. The solution set is (siss C. The solution set is (s|s< D. The solution set is (s|s>

A family has a $98,162, 30-year mortgage at 6.3% compounded monthly. Find the monthly payment. Also find the unpaid balance after the following periods of time.. (A) 10 years (B) 20 years (C) 25 years

Show that Z is integrally closed. and more generally, show that a UFD is integrally closed.

Find the partial sum S12 for the arithmetic sequence with a = 6, d= 7. S12

Determine whether each sequence is arithmetic, geometric, or neither. (Write your final answers down here and include your work in your File Upload to receive full credit.) (a) 4, 12, 16, 24,...