To proof: The inequality
Explanation of Solution
Given information:
The inequality
Formula used:
The steps to prove a statement by mathematical induction are as follows:
Step 1. Show that the statement is true for given starting value of n .
Step 2. Assume that the statement is true for
Step 3. Prove that the statement is true for
Proof:
Consider inequality
Recall that the steps to prove a statement by mathematical induction are as follows:
Step 1. Show that the statement is true for given starting value of n .
Step 2. Assume that the statement is true for
Step 3. Prove that the statement is true for
Substitute
The above inequality holds true.
Therefore, the statement is true for
Now, assume that the statement if true for
Therefore, the statement holds true for
Now, prove it for
Rewrite the left hand side of the above inequality as,
From equation (1), it is clear that
Now, it is obvious that
Because if
Result obtained is that
Hence, by the principle of mathematical induction the inequality
Chapter 9 Solutions
EBK PRECALCULUS W/LIMITS
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning