Concept explainers
In Problems 23-27, prove each statement.
is a factor of .
[Hint: ]
To prove: The given statement is a factor of is true for all natural numbers using the Principle of Mathematical Induction.
Answer to Problem 25AYU
As 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 series is a factor of 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
is a factor of -----(1)
Step 1:
Show that statement (1) is true for the natural number .That is which is obviously a factor of . Hence the statement is true for the natural number .
Step 2:
Assume that the statement is true for some natural number .That is is a factor of -----(2)
Step 3:
Prove that the statement is true for the next natural number .That is, to prove tha is a factor of
Consider
In here, the first term is a multiple of , which has a factor and the second term is a multiple of , which obviously has the factor . Therefore is a factor of
As the statement is true for the natural number terms, it is true for all natural numbers by the theorem of mathematical induction.
Chapter 12 Solutions
Precalculus Enhanced with Graphing Utilities
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