...value at simple discount P = S(1 − dt) 4 Future value at compound interest S = P(1 + i)n 5 Present value at compound interest P = S(1 + i)−n 6 Future value of n payments of R at compound rate i (1 + i ) n − 1 i S = Rsn| = R i 7 Present value of n payments of R at compound rate i 1 − (1 + i ) − n i P = Ran| = R i 8 Approximation to bond or debenture yield for given price 1 I + n (C − P ) i ≈ 1 (C + P) 2 9 Present value of an annuity with payments increasing in arithmetic progression P = R[(1 + i)−1 + 2(1 + i)−2 + ... + n(1 + i)−n] (1 + i )a i − n(1 + i ) − n n| = R i 10 Future value of an annuity with payments increasing in arithmetic progression (1 + i )s i − n n| S = R i 11 Present value of an annuity with payments increasing in geometric progression P = R[(1 + i)−1 + (1 + r)(1 + i)−2 + ... + (1 + r)n−1(1 + i)−n] = R(1 + r)−1 a j where j = i − r n| 1 + r 12 Future value of an annuity with payments increasing in geometric progression j S = R(1 + r)n−1 sn| where j = i − r 1 + r...
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...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, and so on. What is the total number of a person’s ancestors in 15 generations. 15. Find the sum of infinite geometric series. (a) [pic] (b) [pic] 16. The nth term of an arithmetic progression is [pic]. Find the first two terms and also the common difference of the arithmetic progression. 17. (i) Find the sum of the positive integers less than 100 which are multiples of 3 or...
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...ALGEBRA - Tips and Hints * Level 1 * Level 2 * Level 3 Browse Topics * 1. Quadratic Equations * * 1.1. Discriminant * 1.2. Complex Solutions? * 1.3. Remainder Theorem * 1.4. Factor theorem * 2. Functions * * 2.1. Defining the Graph of a Function * 2.2. Some of the important functions * 2.3. Even and Odd Functions * 3. Inequalities * * 3.1. Solving Inequalities * 3.2. Method to Solve Linear, Polynomial or Absolute Value Inequalities * 3.3. Method to Solve Rational Inequalities * 4. Progressions * * 4.1. Series * 4.2. Arithmetic progressions * 4.3. The sum of an arithmetic series * 4.4. Geometric progressions * 4.5. The Sum of a Geometric Series * 4.6. Convergence of geometric series 1. Quadratic Equations A quadratic equation is a second-order polynomial equation in a single variable x. ax2+bx+c=0 -------- (1) with a≠0 Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has two solutions. These solutions may be both real or complex. The roots x can be found by completing the square, x2+bax=−ca -------- (2) (x+b2a)2=−ca+b4a2=b2−4ac4a2 -------- (3) x+b2a=±b2−4ac2a -------- (4) Solving for x then gives x=−b±b2−4ac2a -------- (5) This equation is known as the quadratic formula. The letters a, b and c are coefficients (we know those values). They can have any...
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...groceries before checkout, totaling up the monthly bills, estimating the distance and time a car ride is to a place a person has not been. The problems worked this week have showed how math works in the real world. This paper will show how two math problems from chapter five real world applications numbers 35 and 37 worked out. Number 35 A person hired a firm to build a CB radio tower. The firm charges $100 for labor for the first 10 feet. After that, the cost of labor for each succeeding 10 feet is $25 more than the preceding 10 feet. That is, the nest 10 feet will cost $125; the next 10 feet will cost $150, etc. How much will it cost to build a 90-foot tower? Solving this problem involves the arithmetic sequence. The arithmetic sequence is a sequence of numbers in which each succeeding term differs from the preceding term by the same amount (Bluman, 2011). n = number of terms altogether n = 9 d = the common differences d = 25 ª1 = first term ª1 = 100 ªn = last term ª2 = ª9 The formula used to solve this problem came from the book page 222. ªn = ª1 + (n -1)d ª9 = 100 + (9-1)25 ª9 = 100 + (8)25 ª9 = 100 + 200 ª9 = 300 To solve for...
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...pattern of adding a fixed amount from one term to the next, it is referred to as an arithmetic sequence. The number added to each term is constant (always the same). The fixed amount is called the common difference, d,referring to the fact that the difference between two successive terms yields the constant value that was added. To find the common difference, subtract the first term from the second term. 1. Find the common difference for this arithmetic sequence 5, 9, 13, 17 ... | 1. The common difference, d, can be found by subtracting the first term from the second term, which in this problem yields 4. Checking shows that 4 is the difference between all of the entries. | 2. Find the common difference for the arithmetic sequence whose formula is an = 6n + 3 | 2. The formula indicates that 6 is the value being added (with increasing multiples) as the terms increase. A listing of the terms will also show what is happening in the sequence (start with n = 1). 9, 15, 21, 27, 33, ... The list shows the common difference to be 6. | 3. Find the 10th term of the sequence 3, 5, 7, 9, ... | 3. n = 10; a1 = 3, d = 2 The tenth term is 21. | 4. Find a7 for an arithmetic sequence where a1 = 3x and d = -x. | 4. n = 7; a1 = 3x, d = -x | 5. Find t15 for an arithmetic sequence where t3 = -4 + 5i and t6 = -13 + 11i | 5. Notice...
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...A Free Exam for 2006-15 VCE study design Engage Education Foundation VCE Revision Seminars As a not for profit, this free exam was made possible by our VCE Exam Revision Seminars. Sept 19 - Oct 18 2015. • 24 different subjects • VCAA Assessors • Huge set of notes, teacher slides and an exam • 6.5hrs all located at the University of Melbourne Visit http://ee.org.au/enrol to enrol now! Units 3 and 4 Further Maths: Exam 1 Practice Exam Question and Answer Booklet Duration: 15 minutes reading time, 1 hour 30 minutes writing time Structure of book: Section A B Number of questions 13 54 Number of questions to be answered 13 27 Total Number of Modules Number of modules to be answered 6 3 Number of marks 13 27 40 Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers and rulers. Students are not permitted to bring into the examination room: blank sheets of paper and/or white out liquid/tape. No calculator is allowed in this examination. Materials supplied: This question and answer booklet of 25 pages. Instructions: You must complete all questions of the examination. Write all your answers in the spaces provided in this booklet. Units 3 and 4 Further Maths: Exam 1: Free Exam A The Engage Education Foundation Section A – Multiple-choice questions Instructions Answer all questions by circling your choice. Choose the response that is correct...
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...Died: 29-Dec-1834 Location of death: St. Catherine, near Bath, England Cause of death: unspecified Remains: Buried, Bath Abbey, Bath, England Gender: Male Race or Ethnicity: White Occupation: Economist Nationality: England What many know, at least those with an elementary knowledge of economics or politics, is that Malthus is the surname of a man, who, a couple of hundred years back, said that man, sooner or later, universally, will run up against himself; that the population of mankind will eventually outstrip man's ability to supply himself with the necessities of life. The Malthusian doctrine, as stated in "Essay on the Principle of Population," was expressed as follows: "population increases in a geometric ratio, while the means of subsistence increases in an arithmetic ratio." Well, that seems plain enough, and perfectly understandable, if there is too many people and not enough food, then, certainly, there is going to be problems. Malthus developed his theory, at least to this extent: that left alone, no matter all the problems short of worldwide catastrophe, humankind will survive, as, nature has a natural way to cut population levels: "crime, disease, war, and vice," being, the necessary checks on population." This proposition, as was made by Malthus in 1798, was to cause quite a public stir, then, and yet today. The English economist Thomas Robert Malthus, b. Feb. 14, 1766, d. Dec. 29, 1834, was one of the earliest thinkers to study population growth as it relates...
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... 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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...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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...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) | -2 1/2 2 -1/2 | 10. What name do we give a sequence with an unlimited number of terms? (Points : 3) | Finite series Infinite series Infinite...
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...F.A.M.I.L.Y. (Hook) F.A.M.I.L.Y. What does it mean? F.A.M.I.L.Y. Who they supposed to be? Fake Ass Mutherfuckers I Love You Failing At Making Intelligent Logical Youth 1. Fake Ass Mutherfuckers I Love You 2. Forget All Meaningless I Love You’s 3. Failing A Million infinite long Years 4. Frequently Ask M I L Y 5. Forever and Miserable I L Y 6. Fight And Make-up In lost Years 7. Fight Apologize Make-up Imitating Little Young 8. Forming/form A Multitude in lost year 9. Finding A Moment In Life Year 10. Forgot about Madly In Love Yeah-yeah 11. Fight after missing in love yeah-yeah Verse 1 Take my words as verbal abuse because these are things you can’t define For where were the grandparents that created my Leviathans Grandparents are like the first beginning of the entity Without the beginning there is no origination to learn from Take the Keynesian view, aggregate demand does not necessarily Equal the productive capacity of the economy; Instead, it is influenced by a host of factors And sometimes behaves erratically, Affecting production, employment, and the biggest part A Generation, for what does a F.A.M.I.L.Y. mean if you can’t theoretically explain it? Verse 2 I’m hearing the wedding bells, all of a sudden that ship sails They took vows, it was never carry out, What started as intimacy turn to intimidating And the wedding ring on their finger don’t signify a thing Nor does that marriage certificate...
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...BACHELOR OF BUSINESS (HONS) IN INTERNATIONAL BUSINESS Title : GROUP ASSIGNMENT Submission Date : 21/11/2014 Lecturer : Ms.Hannah Title : GROUP ASSIGNMENT Submission Date : 21/11/2014 Lecturer : Ms.Hannah BBI1214 / BBI1213 ● BUSINESS MATHEMATICS Name : YAZID MUHAMMAD BAJAMAL 110041112 WADAH MOHAMED HAMAD ELNEEL 110039251 MOHAMMED SALEH BAHUBAISHI 110037771 UNAL KAVASTAN 110037858 MONIRUL HASAN 110040569 Semester : 1 Academic Honesty Policy Statement I, hereby attest that contents of this attachment are my own work. Referenced works, articles, art, programs, papers or parts thereof are acknowledged at the end of this paper. This includes data excerpted from CD-ROMs, the Internet, other private networks, and other people’s disk of the computer system. Student’s Signature : _____________________________ SUPERVISOR’S COMMMENTS/GRADE: | for office use only DATE : ______________ TIME : ________________ RECEIVER’S NAME : _____ | Assigment 1 ● Question 1 ● An Invoice of RM 45.000 dated 28/03/2014 with cash discount terms of 2/15, 1/40, n/60. What would be the discount amount if : I. Payment is made on the 10/04/2014 2100x RM 45,000 =RM 900 Discount amount : RM 45,000 – RM 900 = RM 44,100 II. Payment is made on the 1/05/2014 1100 x RM 45,000 =RM 450 Discount amount : RM 45,000 – RM 450 ...
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...Score: ______ / ______ Name: _Kennedy Maddox Student Number: ______________________ | | |1.Elsie is making a quilt using quilt blocks like the one in the diagram. | | | |[pic] | | | |a. How many lines of symmetry are there?Type your answer below. | | | |4 lines of symmetry | | | | ...
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...1. Sum of n terms in a Geometric Sequence Sn= a1 (1-rn )1-r to find r divide a2a1 2. Summation Notation i through n example: i=1nai = a1 + a2 + a3……. an 3. Example: 10i5i= 10*9*8*7*6*5i5i= 30240 4. Finding n terms in arithmetic sequence an=a1+ (n-1)d 5. Finding n terms in geometric sequence an=a1*rn-1 6. Sum of first n terms of an arithmetic sequence Sn=(a1+an)*n2 7. Absolute value of a Complex number Z=a2+b2 a + bi 8. Finding polar form (z= r*(cos theta + i* sin theta) of complex number (z= a + bi) A = r*cos theta B= r* sin theta R=a2+b2 Theta = tan-1 (ba ) 9. Converting from rectangular (x,y) to polar (r,theta) R= x2+ y2 theta = tan-1 (yx) 10. Dick’s theorem for finding roots (a+bi)^ R=a2+ b2 theta = tan-1 (ba ) r^*( cos (theta * ^) + i* sin (theta * ^)) 11. Finding rectangular from polar X= r*cos theta Y=r*sin theta 12. Law of Sines Sin Aa=Sin Bb= Sin CC 13. Angle of oblique triangle Area = b*c*Sin A2 a*b*Sin C2 a*c*Sin B2 14. Law of Cosines a= b2+c2- 2*b*c*Cos A b= a2+c2- 2*a*c*Cos B c= a2+b2- 2*a*b* Cos C 15. Heroin angle Area = S*S-a*S-b*(S-c) S = a+b+c2 16. Permutations nPr= n!n-r! 17. Combinations nCr= n!n*r!*r! 18. Magnitude of v v= a2+b2 19. Position vector Initial point (x1,y1) Terminal point (x2,y2) V = x2-x1*i+y2- y1*j 20. Word problem ...
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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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