Math 365 Review: Problems on Arithmetic Sequences, Real Numbers, and Algebra - Prof. Sherr, Study notes of Mathematics

Download Math 365 Review: Problems on Arithmetic Sequences, Real Numbers, and Algebra - Prof. Sherr and more Study notes Mathematics in PDF only on Docsity! © Scarborough, November 2011, Math 365 Review 1 1. Negate, “All apples are red or some pears are yellow.” 2. Find the sum of the first 21 terms of an arithmetic sequence in which the eighteenth term is 31 117 5 and the thirty-third term is 5222331  . 3. Of the four main operations on real numbers, which ones are commutative? 4. GCD (2332, 1590) = 5. Prove or disprove: If , , ,a b c d   and , , 0b c d  , then a b c d is a rational number. 6. 2 219 5 4 360 18 2       © Scarborough, November 2011, Math 365 Review 2 7. Using the definition of less than, show 3 29 4    . 8. Fully simplify 2 1 8 4 14 4 5 x y z y z  . 9. If U = {a, b, c, d, e}, F = {b, c}, G = {d}, and H = {c, d, e}, find the following. a. F G b. ( )n U c. G H d. ( )n G H e. How many proper subsets does H have? List one non-empty proper subset of H. 10. Over the rational numbers, factor completely: 848 243a  11. What is the contrapositive of the statement, “A whippet is a dog.” 12. Fully simplify:     3 2 3 5 1 0 4 5 2 x yz x y z    © Scarborough, November 2011, Math 365 Review 5 25. Which sets, that we have studied, are closed under the four main operations? 26. What is the Fundamental Counting Principle? Be able to apply it. 27. What are the relationships among the operations of addition, subtraction, multiplication and division? 28.  4 948502 16237582024804 zyx 29. What are the properties of the additive inverse? 30. If f(x) = 3x – 7 and g(x) = 6x + 9, find and simplify fg gf o 32  . What is its domain? 31. Give an example of a relation that is transitive, but not reflexive nor symmetric. 32. Give an example of an equivalence relation. 33. Give an example of the Division Algorithm and why it is needed. 34. Which of the following numbers divide 8415? 2 3 4 5 6 8 9 10 11 35. Prime factor 3864. How many positive divisors does 3864 19 have? 36. What is the largest prime you would need to check to see if 517 is prime? © Scarborough, November 2011, Math 365 Review 6 37. What is the lcm(36,54)? 38. What is the gcd(24,50)? 39. Find 3 rational numbers between 5 17 and 054.3 , and order them from least to greatest. 40. Prove: If a b and c d are rational numbers, then a c ad bc b d bd    . 41. Give an example of a ratio of integers that can be represented by a non-terminating decimal. 42. 4.56  3.27  1.48 = 43. If y is proportional to x such that y = 10 when x = 9, find the constant of proportionality. 44. On a map the scale is 2 inches to 100 miles. If two cities are actually 65 miles apart, how far apart will they be on the map? 45. It takes Devon 3 hours longer than Molly to paint a kitchen. If they work together, it takes 36 7 hours to complete the job. How long does it take Devon to paint the kitchen by himself? 46. 1.45 is what percent? 47. What percent of 30 is 20? 48. Fifty dollars is the original cost of a pair of shoes. The shoes are 20% off the original price, and with a coupon you can get an additional 8% off. What is your total percent discount? How much will you pay for the shoes if the sales tax is 5%? 49. Give the largest 3-digit number that has exactly 3 positive divisors. 50. The sum of three consecutive odd numbers is 93. What is the largest of the three numbers? © Scarborough, November 2011, Math 365 Review 7 51. Justify each step. (27.1)(3.9) a. = (2 * 10 1 + 7 * 10 0 + 1 * 10 -1 )(3 * 10 0 + 9 * 10 -1 ) b. = 2 * 10 1 * 3 * 10 0 + 2 * 10 1 * 9 * 10 -1 + 7 * 10 0 * 3 * 10 0 + 7 * 10 0 * 9 * 10 -1 + 1 * 10 -1 *3 * 10 0 + 1 * 10 -1 * 9 * 10 -1 c. = 2 * 3 * 10 1 * 10 0 + 2 * 9 * 10 1 * 10 -1 + 7 * 3 * 10 0 * 10 0 + 7 * 9 * 10 0 * 10 -1 + 1 * 3 * 10 -1 *10 0 + 1 * 9 * 10 -1 * 10 -1 d. = 6 * 10 1 * 10 0 + 18 * 10 1 * 10 -1 + 21 * 10 0 * 10 0 + 63 * 10 0 * 10 -1 + 1 * 3 * 10 -1 *10 0 + 1 * 9 * 10 -1 * 10 -1 e. = 6 * 10 1 * 10 0 + 18 * 10 1 * 10 -1 + 21 * 10 0 * 10 0 + 63 * 10 0 * 10 -1 + 3 * 10 -1 *10 0 + 9 * 10 -1 * 10 -1 f. = 6 * 10 1 + 18 * 10 0 + 21 * 10 0 + 63 * 10 -1 + 3 * 10 -1 + 9 * 10 -2 g. = 6 * 10 1 + (1 * 10 1 + 8 * 10 0 ) * 10 0 + (2 * 10 1 + 1 * 10 0 ) * 10 0 + (6 * 10 1 + 3 * 10 0 ) * 10 -1 + 3 * 10 -1 + 9 * 10 -2 h. = 6 * 10 1 + 1 * 10 1 * 10 0 + 8 * 10 0 * 10 0 + 2 * 10 1 * 10 0 + 1 * 10 0 * 10 0 + 6 * 10 1 * 10 -1 + 3 * 10 0 * 10 -1 + 3 * 10 -1 + 9 * 10 -2 i. = 6 * 10 1 + 1 * 10 1 + 8 * 10 0 + 2 * 10 1 + 1 * 10 0 + 6 * 10 0 + 3 * 10 -1 + 3 * 10 -1 + 9 * 10 -2 j. = 6 * 10 1 + 1 * 10 1 + 2 * 10 1 + 8 * 10 0 + 1 * 10 0 + 6 * 10 0 + 3 * 10 -1 + 3 * 10 -1 + 9 * 10 -2 k. = (6 + 1) * 10 1 + 2 * 10 1 + (8 + 1 + 6) * 10 0 + (3 + 3) * 10 -1 + 9 * 10 -2 l. = 7 * 10 1 + 2 * 10 1 + 15 * 10 0 + 6 * 10 -1 + 9 * 10 -2 m. = 7 * 10 1 + 2 * 10 1 + (1 * 10 1 + 5 * 10 0 ) * 10 0 + 6 * 10 -1 + 9 * 10 -2 n. = 7 * 10 1 + 2 * 10 1 + 1 * 10 1 * 10 0 + 5 * 10 0 * 10 0 + 6 * 10 -1 + 9 * 10 -2 o. = 7 * 10 1 + 2 * 10 1 + 1 * 10 1 + 5 * 10 0 + 6 * 10 -1 + 9 * 10 -2 p. = (7 + 2 + 1) * 10 1 + 5 * 10 0 + 6 * 10 -1 + 9 * 10 -2 q. = 10 * 10 1 + 5 * 10 0 + 6 * 10 -1 + 9 * 10 -2 r. = (1 * 10 1 + 0 * 10 0 ) * 10 1 + 5 * 10 0 + 6 * 10 -1 + 9 * 10 -2 s. = 1 * 10 1 * 10 1 + 0 * 10 0 * 10 1 + 5 * 10 0 + 6 * 10 -1 + 9 * 10 -2 t. = 1 * 10 2 + 0 * 10 1 + 5 * 10 0 + 6 * 10 -1 + 9 * 10 -2 u. = 105.69