Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
9. Sequences, Series, & Induction
Sequences
4:08 minutes
Problem 25b
Textbook Question
Textbook QuestionIn Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n. 2 is a factor of n^2 - n.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Mathematical Induction
Mathematical induction is a proof technique used to establish the truth of a statement for all positive integers. It involves two main steps: the base case, where the statement is verified for the initial value (usually n=1), and the inductive step, where one assumes the statement holds for an arbitrary positive integer k and then proves it for k+1. This method effectively demonstrates that if the statement is true for one integer, it must be true for all subsequent integers.
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Factors and Divisibility
A factor of a number is an integer that divides that number without leaving a remainder. In the context of the given statement, we are interested in showing that 2 is a factor of the expression n^2 - n for all positive integers n. This involves understanding how to manipulate algebraic expressions and apply the definition of divisibility to confirm that the result is indeed an even number, which is divisible by 2.
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Algebraic Manipulation
Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions using various mathematical operations. In this problem, one must factor the expression n^2 - n to show that it can be expressed in a form that clearly reveals its divisibility by 2. Recognizing that n^2 - n can be factored as n(n - 1) helps in understanding that the product of two consecutive integers (n and n-1) is always even, thus confirming that 2 is a factor.
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