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Concept explainers
Use mathematical induction to prove that if , then
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To prove: The given statement if is true for all natural numbers using the Principle of Mathematical Induction.
Answer to Problem 30AYU
The statement is true for the natural number terms, it is true for all natural numbers by the theorem of mathematical induction.
Explanation of Solution
Given:
Statements says the
Formula used:
The Principle of Mathematical Induction
Suppose that the following two conditions are satisfied with regard to a statement about natural numbers:
CONDITION I: The statement is true for the natural number 1.
CONDITION II: If the statement is true for some natural number
Proof:
Consider the statement
Step 1: Show that statement (1) is true for the natural number
That is
Step 2: Assume that the statement is true for some natural number
That is
Step 3: Prove that the statement is true for the next natural number
That is, to prove that
Consider
As the statement is true for the natural number
Chapter 12 Solutions
Precalculus Enhanced with Graphing Utilities
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