Patterns & Inductive Reasoning UNIT 2 Patterns & Inductive Reasoning

Inductive reasoning Reasoning that is based on patterns you observe. If you observe a pattern in a sequence, you can use inductive reasoning to tell what the next terms in the sequence will be.

Describing a Visual Pattern Sketch the next figure in the pattern. 4 5 1 2 3

Ex. 1: Describing a Visual Pattern - Solution The sixth figure in the pattern has 6 squares in the bottom row. 5 6

Ex. 2: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. 1, 4, 16, 64 Many times in number pattern, it is easiest listing the numbers vertically rather than horizontally.

Ex. 2: Describing a Number Pattern How do you get to the next number? That’s right. Each number is 4 times the previous number. So, the next number is 256, right!!! Describe a pattern in the sequence of numbers. Predict the next number. 1 4 16 64

Ex. 2: Describing a Number Pattern How do you get to the next number? That’s right. You add 3 to get to the next number, then 6, then 9. To find the fifth number, you add another multiple of 3 which is +12 or 25, That’s right!!! Describe a pattern in the sequence of numbers. Predict the next number. -5 -2 4 13

Goal 2: Using Inductive Reasoning A conjecture is an unproven statement that is based on observations. Verify the conjecture. Use logical reasoning to verify the conjecture is true IN ALL CASES.

Ex. 3: Making a Conjecture Complete the conjecture. Conjecture: The sum of the first n odd positive integers is ?. How to proceed: List some specific examples and look for a pattern.

Ex. 3: Making a Conjecture First odd positive integer: 1 = 12 1 + 3 = 4 = 22 1 + 3 + 5 = 9 = 32 1 + 3 + 5 + 7 = 16 = 42 The sum of the first n odd positive integers is n2.

Counterexample To prove that a conjecture is true, you need to prove it is true in all cases. To prove that a conjecture is false, you need to provide a single counter example. A counterexample is an example that shows a conjecture is false.

Ex. 4: Finding a counterexample Show the conjecture is false by finding a counterexample. Conjecture: For all real numbers x, the expressions x2 is greater than or equal to x.

Ex. 4: Finding a counterexample- Solution Conjecture: For all real numbers x, the expressions x2 is greater than or equal to x. The conjecture is false. Here is a counterexample: (0.5)2 = 0.25, and 0.25 is NOT greater than or equal to 0.5. In fact, any number between 0 and 1 is a counterexample.

Note: Not every conjecture is known to be true or false. Conjectures that are not known to be true or false are called unproven or undecided.

Ex. 5: Using Inductive Reasoning in Real-Life Moon cycles. A full moon occurs when the moon is on the opposite side of Earth from the sun. During a full moon, the moon appears as a complete circle.

Ex. 6: Using Inductive Reasoning in Real-Life Use inductive reasoning and the information below to make a conjecture about how often a full moon occurs. Specific cases: In 2017, the first six full moons occur on January 12, February 10, March 12, April 11, May 10 and June 9.

Ex. 6: Using Inductive Reasoning in Real-Life - Solution A full moon occurs every 29 or 30 days. This conjecture is true. The moon revolves around the Earth approximately every 29.5 days. Inductive reasoning is very important to the study of mathematics. You look for a pattern in specific cases and then you write a conjecture that you think describes the general case. Remember, though, that just because something is true for several specific cases does not prove that it is true in general.

Ex. 6: Using Inductive Reasoning in Real-Life - Solution A full moon occurs every 29 or 30 days. This conjecture is true. The moon revolves around the Earth approximately every 29.5 days. Inductive reasoning is very important to the study of mathematics. You look for a pattern in specific cases and then you write a conjecture that you think describes the general case. Remember, though, that just because something is true for several specific cases does not prove that it is true in general.