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given quantity. (See Example 2.) 7. the sum of an even integer and an odd integer

given quantity. (See Example 2.) 7. the sum of an even integer and an odd integer

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Exercises and Explanations: Patterns, Conjectures, and Logic** ### Pattern Recognition and Prediction 1-6. Describe the pattern, then write or draw the next two numbers, letters, or figures. - **Examples**: - 1, -2, -3, -4, ... - 2, 0, 0, 6, 12, 20, ... - X, Y, X, W… - A, D, G, J, M... Figures: Square, Pentagon, Hexagon, Cubes. ### Conjecture Making and Testing 7-10. Make and test a conjecture about the given quantity. - Sum of an even and an odd integer. - Product of any two even integers. - Quotient of a number and its reciprocal. - Quotient of two negative integers. ### Counterexamples 11-14. Find a counterexample to show that the conjecture is false. - Product of two positive numbers is always greater than either number. - n is a nonzero integer, then \( n + \frac{1}{n} \) is greater than 1. - If two angles are supplements, each one of the angles must be acute. - A line divides segment \(\overline{MN}\) into line segments. So, the line is a segment bisector of \(\overline{MN}\). ### Using the Law of Detachment 15-18. Deduce what you can conclude from the given information. - Downloading a GIF will crash your device. ### Using the Law of Syllogism 19-22. Write a new conditional statement from two true statements. - Example: If \( x = -2 \), then \(|x| > 2\). ### Law of Logic Illustration 23-26. State the law of logic that is illustrated. - Example: If you do your homework, you can watch TV. ### Inductive Reasoning 27-28. Make a conjecture about the given quantity using inductive reasoning. - Example: Sum of the first n odd integers. This educational resource is designed to help students develop their skills in pattern recognition, making conjectures, and logical reasoning.
Transcribed Image Text:**Exercises and Explanations: Patterns, Conjectures, and Logic** ### Pattern Recognition and Prediction 1-6. Describe the pattern, then write or draw the next two numbers, letters, or figures. - **Examples**: - 1, -2, -3, -4, ... - 2, 0, 0, 6, 12, 20, ... - X, Y, X, W… - A, D, G, J, M... Figures: Square, Pentagon, Hexagon, Cubes. ### Conjecture Making and Testing 7-10. Make and test a conjecture about the given quantity. - Sum of an even and an odd integer. - Product of any two even integers. - Quotient of a number and its reciprocal. - Quotient of two negative integers. ### Counterexamples 11-14. Find a counterexample to show that the conjecture is false. - Product of two positive numbers is always greater than either number. - n is a nonzero integer, then \( n + \frac{1}{n} \) is greater than 1. - If two angles are supplements, each one of the angles must be acute. - A line divides segment \(\overline{MN}\) into line segments. So, the line is a segment bisector of \(\overline{MN}\). ### Using the Law of Detachment 15-18. Deduce what you can conclude from the given information. - Downloading a GIF will crash your device. ### Using the Law of Syllogism 19-22. Write a new conditional statement from two true statements. - Example: If \( x = -2 \), then \(|x| > 2\). ### Law of Logic Illustration 23-26. State the law of logic that is illustrated. - Example: If you do your homework, you can watch TV. ### Inductive Reasoning 27-28. Make a conjecture about the given quantity using inductive reasoning. - Example: Sum of the first n odd integers. This educational resource is designed to help students develop their skills in pattern recognition, making conjectures, and logical reasoning.
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