Concept explainers
To prove: The given statement for is true for all natural numbers using the Principle of Mathematical Induction.
Answer to Problem 27AYU
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 for 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
for -----(1)
Step 1: Show that statement (1) is true for the natural number .
That is . Hence the statement is true for the natural number .
Step 2: Assume that the statement is true for some natural number .
That is for -----(2)
Step 3: Prove that the statement is true for the next natural number .
That is, to prove that , for
Consider
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
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