...Write the first four terms of the sequence whose general term is an = 2( 4n - 1) (Points : 3) | 6, 14, 22, 30 -2, 6, 14, 22 3, 7, 11, 15 6, 12, 18, 24 | 2. Write the first four terms of the sequence an = 3 an-1+1 for n ≥2, where a1=5 (Points : 3) | 5, 15, 45, 135 5, 16, 49, 148 5, 16, 46, 136 5, 14, 41, 122 | 3. Write a formula for the general term (the nth term) of the arithmetic sequence 13, 6, -1, -8, . . .. Then find the 20th term. (Points : 3) | an = -7n+20; a20 = -120 an = -6n+20; a20 = -100 an = -7n+20; a20 = -140 an = -6n+20; a20 = -100 | 4. Construct a series using the following notation: (Points : 3) | 6 + 10 + 14 + 18 -3 + 0 + 3 + 6 1 + 5 + 9 + 13 9 + 13 + 17 + 21 | 5. Evaluate the sum: (Points : 3) | 7 16 23 40 | 6. Find the 16th term of the arithmetic sequence 4, 8, 12, .... (Points : 3) | -48 56 60 64 | 7. Identify the expression for the following summation:(Points : 3) | 6 3 k 4k - 3 | 8. A man earned $2500 the first year he worked. If he received a raise of $600 at the end of each year, what was his salary during the 10th year? (Points : 3) | $7900 $7300 $8500 $6700 | 9. Find the common ratio for the geometric sequence.: 8, 4, 2, 1, 1/2 (Points : 3)...
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...How satisfied are you with Wikipedia? Your feedback is important to us! As a token of appreciation for your support you get a chance of winning a Wikipedia T-shirt. Click here to learn more! Calculus From Wikipedia, the free encyclopedia Jump to: navigation, search This article is about the branch of mathematics. For other uses, see Calculus (disambiguation). | It has been suggested that Infinitesimal calculus be merged into this article or section. (Discuss) Proposed since May 2011. | Topics in Calculus | Fundamental theorem Limits of functions Continuity Mean value theorem [show]Differential calculus | Derivative Change of variables Implicit differentiation Taylor's theorem Related rates Rules and identities:Power rule, Product rule, Quotient rule, Chain rule | [show]Integral calculus | IntegralLists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order | [show]Vector calculus | Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem | [show]Multivariable calculus | Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian | | Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes...
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...Arithmetic and geometric progressions mcTY-apgp-2009-1 This unit introduces sequences and series, and gives some simple examples of each. It also explores particular types of sequence known as arithmetic progressions (APs) and geometric progressions (GPs), and the corresponding series. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: • recognise the difference between a sequence and a series; • recognise an arithmetic progression; • find the n-th term of an arithmetic progression; • find the sum of an arithmetic series; • recognise a geometric progression; • find the n-th term of a geometric progression; • find the sum of a geometric series; • find the sum to infinity of a geometric series with common ratio |r| < 1. Contents 1. Sequences 2. Series 3. Arithmetic progressions 4. The sum of an arithmetic series 5. Geometric progressions 6. The sum of a geometric series 7. Convergence of geometric series www.mathcentre.ac.uk 1 c mathcentre 2009 2 3 4 5 8 9 12 1. Sequences What is a sequence? It is a set of numbers which are written in some particular order. For example, take the numbers 1, 3, 5, 7, 9, . . . . Here, we seem to have a rule. We have a sequence of odd numbers. To put this another way, we start with the number 1, which is an odd number, and then each successive number is obtained...
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...MCR 3U Exam Review Unit 1 1. Evaluate each of the following. a) b) c) 2. Simplify. Express each answer with positive exponents. a) b) c) 3. Simplify and state restrictions a) b) c) d) 4. Is Justify your response. 5. Is Justify your response. Unit 2 1. Simplify each of the following. a) b) c) 2. Solve. a) b) c) 3. Solve. Express solutions in simplest radical form. a) b) 4. Find the maximum or minimum value of the function and the value of x when it occurs. a) b) 5. Write a quadratic equation, in standard form, with the roots a) and and that passes through the point (3, 1). b) and and that passes through the point (-1, 4). 6. The sum of two numbers is 20. What is the least possible sum of their squares? 7. Two numbers have a sum of 22 and their product is 103. What are the numbers ,in simplest radical form. Unit 3 1. Determine which of the following equations represent functions. Explain. Include a graph. a) b) c) d) 2. State the domain and range for each relation in question 1. 3. If and , determine the following: a) b) 4. Let . Determine the values of x for which a) b) Recall the base graphs. 5. Graph . State the domain and range. Describe...
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...Life, Death, and the Critical Transition: Finding Liveness Bugs in Systems Code Charles Killian, James W. Anderson, Ranjit Jhala, and Amin Vahdat University of California San Diego {ckillian,jwanderson,jhala,vahdat}@cs.ucsd.edu Abstract finding bugs with model checking currently requires the programmer to have intimate knowledge of the low-level Modern software model checkers find safety violations: actions or conditions that could result in system failure. breaches where the system has entered some bad state. For We contend that for complex systems the desirable bemany environments however, particularly complex con- haviors of the system may be specified more easily than current and distributed systems, we argue that liveness identifying everything that could go wrong. Of course, properties are both more natural to specify and more im- specifying both desirable conditions and safety assertions portant to check. Liveness conditions specify desirable is valuable; however, current model checkers do not have system conditions in the limit, with the expectation that any mechanism for verifying whether desirable system they will be temporarily violated, perhaps as a result of properties can be achieved. Examples of such properties failure or during system initialization. include: i) a reliable transport eventually delivers all mesExisting software model checkers cannot verify live- sages even in the face of network losses and delays, ii) all ness because doing so...
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...rate 2. Figure 1 – A binary information signal To modulate this signal, we would multiply this sequence with a sinusoid and its spectrum would look like as In figure 2. The main lobe of its spectrum is 2 Hz wide. The larger the symbol rate the larger the bandwidth of the signal. Figure 2 – Spectrum of a binary signal of rate 2 bps Now we take an another binary sequence of data rate 8 times larger than of sequence shown in Fig. 1. Copyright 2002 Charan Langton www.complextoreal.com CDMA Tutorial 2 Figure 3 – A new binary sequence which will be used to modulate the information sequence Instead of modulating with a sinusoid, we will modulate the sequence 1 with this new binary sequence which we will call the code sequence for sequence 1. The resulting signal looks like Fig. 4. Since the bit rate is larger now, we can guess that the spectrum of this sequence will have a larger main lobe. Figure 4 – Each bit of sequence 1 is replaced by the code sequence The spectrum of this signal has now spread over a larger bandwidth. The main lobe bandwidth is 16 Hz instead of 2 Hz it was before spreading. The process of multiplying the information sequence with the code sequence has caused the information sequence to inherit the spectrum of the code sequence (also called the spreading sequence). Figure 5 – The spectrum of the spread signal is as wide as the code sequence The spectrum has spread from 2 Hz to 16 Hz, by a factor of 8. This number is called the the spreading...
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...any student you believe might benefit from them. If you downloaded these notes from a source other than the bit.ly link above, please check there to make sure you are reading the latest version. It may contain additional content and important corrections! April 8, 2016 1 Contents 1 Algebra 1.1 Rules of Basic Operations . . . . . . . . . . . . . . . . . . . . . 1.2 Rules of Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Rules of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Allowed and Disallowed Calculator Functions During the Exam 1.5 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Arithmetic Sequences and Series . . . . . . . . . . . . . . . . . 1.7 Sum of Finite Arithmetic Series (u1 + · · · + un ) . . . . . . . . . 1.8 Partial Sum of Finite Arithmetic Series (uj + · · · + un ) . . . . . 1.9 Geometric Sequences and Series . . . . . . . . . . . . . . . . . . 1.10 Sum of Finite Geometric Series . . . . . . . . . . . . . . . . . . 1.11 Sum of Infinite Geometric Series . . . . . . . . . . . . . . . . . 1.11.1 Example Involving Sum of Infinite Geometric Series . . 1.12 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.1 Sigma Notation for Arithmetic Series . . . . . . . . . . . 1.12.2 Sigma Notation for Geometric Series . . . . . . . . . . . 1.12.3 Sigma Notation for Infinite Geometric Series . . ....
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...Fibonacci Numbers In the 13th century a man named Leonardo of Pisa or Fibonacci founded Fibonacci Numbers. Fibonacci Numbers are “a series of numbers in which each number is the sum of the two preceding numbers” (Burger 57). His book “Liber Abaci” written in 1202 introduced this sequence to Western European mathematics, although they had been described earlier in Indian mathematics. He proved that through spiral counts there is a sequence of numbers with a definite pattern. The simplest series is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…and so on. When looking at this series the pattern proves that adding the previous number to the next will give you the following number in the series. For example, (1+1=2), (2+3=5), and etc. In order to ensure accuracy when using Fibonacci Numbers a formula was created. The formula or rule that follows the Fibonacci sequence is Fn = Fn-1 + Fn-2. By plugging in any numbers in a problem to this equation a student can find the right answer. This gives students the ability to calculate any Fibonacci Number. In modern times society uses these numbers to calculate numerous things. For instance, like the sizes of our arms relative to our torso and even the structure of hurricanes. On another note, Fibonacci Numbers can also be found in patterns in nature. It is truly astonishing to think about how relations in Fibonacci Numbers may possibly be represented in our lives. Works Cited Burger, Edward B., and Michael P. Starbird....
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...will eventually go wrong. I didn't like being the main subject of a conversation when someone fails to report to me on a piece of equipment or system. When I look back on those situations I see I let my confluence flow and should have done something different. When a small compressor starts to fail and its passed its prime and I let the upper chain of command know and they brush it off. I could have used more of my sequence and should have given more details on what my solution is. My favorite thing I dislike the most, is when another person that has been trained to be your back up (in absence) they don't pull the load. I should have used more of my sequence and precise than just leaving it to chance. I liked it when a hot water heater tank needs to be overhauled (recycled) and my boss lets me take the guys on the site to do on the job training. My technical reasoning steps in big time but it has to merge with my sequence and precise patterns. I like being able to break down steam valves and show the how, why, when, and what if. Same as before I use my sequence, technical reasoning and precise to ensure that all things are in place. The most enjoyable part is being able to speak among the higher ranks as a voice for those who's voices can't speak at those meetings. I use all my patterns in those closed door meetings. Sometimes knowing when to be...
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...Score: ______ / ______ Name: ______________________________ Student Number: ______________________ | 1. Elsie is making a quilt using quilt blocks like the one in the diagram. a. How many lines of symmetry are there? Type your answer below. b. Does the quilt square have rotational symmetry? If so, what is the angle of rotation? Type your answers below. | | 2. Solve by simulating the problem. You have a 5-question multiple-choice test. Each question has four choices. You don’t know any of the answers. What is the experimental probability that you will guess exactly three out of five questions correctly? Type your answer below using complete sentences. | | 3. Use the diagram below to answer the following questions. Type your answers below each question. a. Name three points.b. Name four different segments.c. Write two other names for FG.d. Name three different rays. | | 4. Charlie is at a small airfield watching for the approach of a small plane with engine trouble. He sees the plane at an angle of elevation of 32. At the same time, the pilot radios Charlie and reports the plane’s altitude is 1,700 feet. Charlie’s eyes are 5.2 feet from the ground. Draw a sketch of this situation (you do not need to submit the sketch). Find the ground distance from Charlie to the plane. Type your answer below. Explain your work. | | _____ 5. Jason and Kyle both choose a number from 1 to 10 at random. What is the probability that both numbers are odd? Type...
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...5.1.16 ! ROWING! ! PE COURSEWORK D SHAM Section B- AA1- Sculling Strokes - Inconsistent hand heights on recovery! B2 - Technical Model! ! ! ! Elite Technical Model of Performer- Mahe Drysdale! ! In a competitive situation, the consistency of hand heights is important as it is an important factor in determining whether a boat will be sat or not. The most important factor in having a balanced boat through the recovery phase is to have consistent hand heights through the boat. Especially in a competitive situation where the rate per minute is typically high. Mahe Drysdale is a very experienced sculler from New Zealand, who is a current olympic champion with 5 word champion titles in a single scull. Despite Mahe’s age of 34 at that time, he is still a world champion not just because of his experience and amount of training, but also due to the fact that he is very consistent on the water. His consistent blade heights allows him for an early catch, which allows him to take full advantage of a stroke. In the 2012 olympics, Mahe finished first with a close 3/4 length lead to 30 year old Czech Republic single sculler Ondřej Synek. ! ! ! Preparation! To prepare going up the slide, the performer should be sat at the finish with both legs extended, body slightly leaning back. Both oars should be drawn into the chest feathered. ! ! ! Execution! Blade height can only be judged during the recovery phase. Therefore the recovery phase starts with...
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...associated with symbolic representation. 8. What are the aspects to be considered to ensure information transfer between source and destination? Syntactic – Physical form of information Semantic – Meaning Pragmatic – Action taken as a result of interpretation of information 9. State the Markov algorithm It takes the input string X and through a number of steps in the algorithm, transform X to an output string Y 10. Differentiate function & procedure A function is a sub algorithm which is used when one wants a single value returned to the calling routine. A procedure is a sub algorithm which does not return any value explicitly 11. How the character information is represented in memory? A character is represented in memory as a sequence of...
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...TUTORIAL 3 SEQUENCES AND SERIES 3.1 Sequences and Series 1. Find the first four terms and 100th term of the sequence. (a) [pic] (b) [pic] (c) [pic] 2. Find the nth term of a sequence whose first several terms are given. (a) [pic]…… (b) 0, 2, 0, 2, 0, 2 …… 3. Find the sum. (a) [pic] (b) [pic] 4. Write the sum using sigma notation. (a) [pic] (b)[pic] 5. Find the nth term for each of the following sequences. Hence, determine whether the respective sequence is divergent or convergent. For a convergent sequence, state its limits. [pic] [pic] 6. For the sequence [pic]find the nth term and show that the above sequence is convergent and determine its limits. 7. The tenth term of an arithmetic sequence is [pic], and the second term is [pic]. Find the first term. 8. The first term of an arithmetic sequence is 1, and the common difference is 4. Is 11937 a term of this sequence? If so, which term is it? 9. The common ratio in a geometric sequence is [pic], and the fourth term is [pic]. Find the third term. 10. Which term of the geometric sequence 2, 6, 18, … is 118098? 11. Express the repeating decimal as fraction. (a) 0.777… (b) [pic] (c) [pic] 12. Find the sum of the first ten terms of the sequence. [pic] 13. The sum of the first three terms of a geometric series is 52, and the common ratio is r = 3. Find the first term. 14. A person has two parents, four grandparents, eight great-grandparents,...
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...Zubin Panna MA1310 College Mathematics II Module 1 Exercise 1.1 1. Describe an arithmetic sequence in two sentences. A sequence in which each term after the first differs from the preceding term by a constant amount. The difference between consecutive terms is called the common difference of the sequence. 2. Describe a geometric sequence in two sentences. A sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence. 3. Write the first four terms of the sequence. a1= 13 and an = an-1+8 for n ≥ 2 a1 = 13 a2 = an-1+8, (n = 2) a2 = 13 + 8 = 21 a3 = a2 + 8 a3 = 21 + 8 = 29 a4 = a3 + 8 a4 = 29 + 8 = 37 The first four terms of the sequence are 13, 21, 29, and 37 4. Evaluate: 16!2!*14! 16*15*14!2*1*14! = 16*15*14!2*1*14! =16*152 = 2402 =120 5. Find the indicated sum i=15i2 i=15i2 = 12 + 22 + 32 + 42 + 52 = 1 + 4 + 9 + 16 +25 i=15i2 = 55 6. A company offers a starting yearly salary of $33,000 with a raise of $2,500 per year. Find the total salary over a 10-year period. an = a1 + (n - 1)*d, [where n = 10 years; a1 = $33,000; d = $2,500] a10 = 33,000 + (10 - 1) * 2,500 a10 = 33,000 + (9) * 2500 a10 = 33,000 + 22,500 a10 = 55,500 The total salary over a 10-year period will be $55,500. 7. Suppose you have $1 the first day of a month, $5 the second day, $25 the third day, and so on...
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... |TOC: Examine evidence of approved purchase | | | | |and services orders. | |2. |Completeness |All purchases of inventory for the period are |Suppliers’ invoices are numbered using an | | | |recorded. |invoice register or pre numbered vouchers | | | | |and the sequences accounted for. | | | | |TOC: Review the evidence of the accounting | | | | |for numerical sequence of invoices. | |3. |Accuracy |Purchases of inventory are recorded correctly as|Established procedures for review of | | | |to...
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