Concept explainers
In Problems 23-27, prove each statement.
If , then .
To prove: The given statement , then is true for all natural numbers using the Principle of Mathematical Induction.
Answer to Problem 23AYU
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 , then 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
, then -----(1)
Step 1: Show that statement (1) is true for the natural number .
That is which is obvious from the statement. Hence the statement is true for the natural number .
Step 2: Assume that the statement is true for some natural number .
That is , then -----(2)
Step 3: Prove that the statement is true for the next natural number .
That is, to prove that , then
Consider
In here, and . The two numbers which are greater than 1, then their product must be greater than 1.
Hence .
As the statement is true for the natural number terms, it is true for all natural numbers by the theorem of mathematical induction.
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