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1.1 Solving Problems by Inductive Reasoning The development of mathematics can be traced to the Egyptian and Babylonian cultures (3000 B.C. – A.D. 260) as a necessity for problem solving. To solve a problem or perform an operation, a cookbook-like recipe was given, and it was performed repeatedly to solve similar problems. By observing that a specific method worked for a certain type of problem, the Egyptians and the Babylonians concluded that the same method would work for any similar type of problem. Such a conclusion is called a conjecture. A ____________________ is an educated guess based on repeated observations of a particular process or pattern. reasoning ____________________ is characterized by drawing a general conclusion (making a conjecture) from repeated observations of specific examples. The conjecture may or may not be true. In testing a conjecture obtained by inductive reasoning, it takes only one example, called a ________________________, that does not work in order to prove the conjecture false. Inductive reasoning offers a powerful method of drawing conclusions, but there is no guarantee that the observed conjecture will always be true. The conjecture is not accepted as absolute proof until the conjecture is proved using ________________ ____________________ reasoning reasoning . is characterized by applying general principles to specific examples. Ex 1) We refer to 1,2, 3, … as the natural or counting numbers. The general rule for continuing the pattern Ex 2) Use inductive reasoning to determine the probable next number. a) 7, 11, 15, 19, 23 b) 1, 1, 2, 3, 5, 8, 13, 21 c) 3, 9, 27, 81, 243 Cosner - Math 107 Chapter 1 - 1 A ____________________ can be an assumption, law, rule, widely held idea, or observation. Then reason inductively or deductively from the premises to obtain a ____________________. The premises and conclusion make up a ____________________ ____________________. Ex 3) Identify each premise and the conclusion in each of the following arguments. Then tell whether each argument is an example of inductive or deductive reasoning. a) Our house is made of rocks. Both of my next door neighbors have rock houses. Therefore, all houses in our neighborhood are made of rocks. b) All calculators can type the symbol +. I have a calculator. I can type the symbol +. c) Today is Tuesday. Tomorrow will be Wednesday. Cosner - Math 107 Chapter 1 - 2 Ex 4) A list of equations is given. Use the list and inductive reasoning to predict the next equation, and then verify your conjecture. 15873 7 111,111 a) (#34) 15873 14 222,222 15873 21 333,333 15873 28 444,444 1 1 1 2 2 1 1 2 1 2 2 3 3 b) (#42) 1 1 1 3 1 2 2 3 3 4 4 1 1 1 1 4 1 2 2 3 3 4 4 5 5 Ex 5) (#51 page 9) Find a pattern in the figure and use inductive reasoning to predict the next figure. Cosner - Math 107 Chapter 1 - 3 1.2 An Application of Inductive Reasoning: Number Patterns 1.2 Book HW: 31, 33, 51, 52, 54 A ____________________ is an ordered list of numbers. A _________________ _________________ is a list of numbers having a first number, a second number, a third number, and so on, called the ____________________ of the sequence. Arithmetic Sequence: A sequence a , a 1 2 , a 3 , … an , … is called an arithmetic sequence if there is a _________________ _________________ d between successive terms. Geometric Sequence: A sequence a , a 1 2 , a 3 , … an , … is called an geometric sequence if there is a real number r 0 , called the _________________ _________________ between successive terms. Ex 1) Use the method of successive differences to determine the next number in each sequence. d) (#2) 3, 14, 31, 54, 83, 118, … e) (#8) 3, 19, 165, 771, 2503, 6483, 14409 Cosner - Math 107 Chapter 1 - 4 Observe the following pattern: So 1 3 5 7 9 = and 1 3 5 7 9 11 = 1 12 1 3 22 1 3 5 32 1 3 5 7 42 We cannot prove this without a method of proof called mathematical induction. But mathematicians have. So we can generalize this for any counting number n. Ex 2) Several equations are given illustrating a suspected number pattern. Determine what the next equation would be, and verify that it is indeed a true statement. Then find a general formula, if possible. a) 12 13 1 22 13 2 3 1 2 32 13 2 3 33 1 2 3 42 13 2 3 33 4 3 Cosner - Math 107 Chapter 1 - 5 1 2 2 23 1 2 2 b) 3 4 1 2 3 2 45 1 2 3 4 2 1 Triangular Numbers: Ex 3) Identify the eighth triangular number. Cosner - Math 107 Chapter 1 - 6 Ex 4) The first five pentagonal numbers are shown below. Identify the formula to find the nth pentagonal number. Then find the 23rd pentagonal number. Ex 5) (#26) Use the formula S = n2 to find the sum: 1 + 3 + 5 + … + 49 Cosner - Math 107 Chapter 1 - 7 1.3 Strategies for Problem Solving 1.3 Book HW: 2, 15, 27, 40, 58 In the first two sections of this chapter we stressed pattern recognition and the use of inductive reasoning in problem solving. Now we will use probably the most famous study of problem solving techniques developed by George Polya. Polya proposed a four-step method for problem solving. Pólya’s First Principle: Understand the Problem This seems so obvious that it is often not even mentioned, yet students are often hindered in their efforts to solve problems simply because they don’t understand it fully, or even in part. State the problem you are trying to solve in your own words. Clearly write the question(s). Pólya’s Second Principle: Devise a plan There are many reasonable ways to solve problems. he skill at choosing appropriate strategies is best learned by solving many problems. You will find choosing strategies increasingly easy. A partial list of strategies is included: Work backward Make a table or a chart If a formula applies, use it. Look for a pattern Guess and check. Solve a similar simpler problem Use trial and error. Draw a picture Use common sense. Use inductive reasoning Look for a “catch” if an answer seems too obvious or Write an equation and solve it. impossible. Make a list of the strategies (two or more, please). Pólya’s third Principle: Carry out the plan Now is the time to actually solve the problem! The solution should be included. You may run into “dead ends” and unforeseen roadblocks, but be persistent. Pólya’s Fourth Principle: Look back Much can be gained by taking the time to reflect and look back at what you have done; what worked and what didn’t. Doing this will enable you to predict what strategies to use to solve future problems. Rabbit Problem In the year 1202, Fibonacci became interested in the reproduction of rabbits. He created an imaginary set of ideal conditions under which rabbits could breed, and posed the question, "How many pairs of rabbits will there be a year from now?" The ideal set of conditions was a follows: 1) 2) 3) 4) 5) 6) You begin with one male rabbit and one female rabbit. These rabbits have just been born. A rabbit will reach sexual maturity after one month. The gestation period of a rabbit is one month. Once it has reached sexual maturity, a female rabbit will give birth every month. A female rabbit will always give birth to one male rabbit and one female rabbit. Rabbits never die. Ex 1) So how many male/female rabbit pairs are there after one year (12 months)? Step 1: Understand the Problem: Cosner - Math 107 Chapter 1 - 8 Step 2: Devise a Plan Use a chart/table because there is a pattern to how the rabbits will reproduce. Month Number of Pairs at Start Number of New Pairs Produced Number of Pairs at End of Month 1 2 3 4 5 6 7 8 9 10 11 12 Step 3: Carry out the Plan Cosner - Math 107 Chapter 1 - 9 Ex 2) The number of dogs and chickens on a farm add up to 12. The number of legs between them is 28. How many dogs and how many chickens are on the farm if there are at least twice as many chickens as dogs? Use Polya’s four steps. Ex 3) (#10) Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been if she had worked the problem correctly? Ex 4) (#42) How many squares are in the following figure? Cosner - Math 107 Chapter 1 - 10