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In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers .
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To prove: The given statement is true for all natural numbers using the Principle of Mathematical Induction.
Answer to Problem 2AYU
As the statement is true for the natural number terms, hence the statement is true for all natural numbers.
Explanation of Solution
Given:
Statements says the series is true for all natural number.
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 , it is also true for the next natural number . Then the statement is true for all natural numbers.
Proof:
Consider the statement -----(1)
Step 1: Show that statement (1) is true for .
That is Hence the statement is true for natural number .
Step 2: Assume that the statement is true for some natural number .
That is -----(1)
Step 3: Prove that the statement is true for the next natural number .
That is, to prove that
Consider
=
[Substituting equation (1)]
As the statement is true for the natural number terms, hence the statement is true for all natural numbers.
Chapter 12 Solutions
Precalculus Enhanced with Graphing Utilities
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