INDUCTION

Induction

Definition Of Induction

Mathematical Induction is a method generally used to prove or establish that a given statement is true for all natural numbers.

More About Induction

It is a method to prove a proposition which is valid for infinitely many different values of a variable.

Video Examples: Mathematical Induction

Example of Induction

Mathematical induction can be used to prove that 1 + 3 + 5 + ---- + (2n - 1) = n² for all positive integers.
Let P_n be the statement 1 + 3 + 5 + ---- + (2n - 1) = n².
P 1 is true because (2(1) - 1) = 1².
Assume that P_k is true, so that P_k: 1 + 3 + 5 + ------ + (2k - 1) = k². [The Inductive Hypothesis.]
The next term on the left hand side would be [2(k + 1) - 1] = (2k + 1). [Add (2k + 1) on both sides to P_k.]
1 + 3 + 5 + ------ + (2k - 1) + (2k + 1) = k² + (2k + 1) = (k + 1)² = P_k+1
So, the equation is true for n = k + 1.
Therefore, P_n is true for all positive integers, by mathematical induction.

Solved Example on Induction

Ques: Using mathematical induction to prove that 1 + 5 + 9 + ------ + (4n - 3)
= n(2n - 1) for all positive integers.

Solution:

Step 1: Let P_n be the statement 1 + 5 + 9 + ---- + (4n - 3) = n (2n - 1).
Step 2: P 1 is true because (4(1) - 3) = 1(2(1) - 1) [The Anchor.]
Step 3: Assume that P_k is true, so that P_k: 1 + 5 + 9 + ---- + (4k - 3)
= k (2k - 1) [The Inductive Hypothesis.]
Step 4: The next term on the left-hand side would be (4(k + 1) - 3).
Step 5: 1 + 5 + 9 + ---- + (4k - 3) + (4(k + 1) - 3)
= k (2k - 1) + (4(k + 1) - 3)
Step 6: = 2k² - k + 4k + 1
Step 7: = 2k² + 3k + 1
Step 8: = (k + 1)² + k (k + 1)
Step 9: = (k + 1) (2k + 1) [Factor.]
Step 10: = (k + 1) (2(k + 1) - 1)
Step 11: = P_k+1
Step 12: So, the equation is true for n = k + 1.
Step 13: Therefore, Pk is true for all positive integers, by mathematical induction.

Quick Summary

Mathematical induction proves statements for all natural numbers.
It involves a base case, inductive hypothesis, and inductive step.
The base case verifies the statement for the smallest natural number.
The inductive hypothesis assumes the statement is true for some k.
The inductive step proves the statement is true for k+1.

\[ P(n) \implies P(n+1) \]

🍎 Teacher Insights

Emphasize the importance of clearly stating the base case, inductive hypothesis, and inductive step. Use visual aids and concrete examples to illustrate the concept. Encourage students to practice applying mathematical induction to various problems.

🎓 Prerequisites

Basic Algebra
Understanding of Natural Numbers
Propositional Logic

Check Your Knowledge

Q1: Which step is essential in mathematical induction?

A. Proving the statement for n=0 B. Assuming the statement is false for some k C. Proving the base case D. Ignoring the inductive hypothesis

Q2: What is the inductive hypothesis?

A. The statement to be proven B. The assumption that the statement is true for some k C. The step where you prove the statement for k+1 D. The conclusion of the proof

Frequently Asked Questions

Q: What is the purpose of the base case?
A: The base case establishes the foundation upon which the inductive argument is built. Without it, the subsequent steps are meaningless.

Q: What is the inductive hypothesis?
A: The inductive hypothesis assumes that the statement is true for some arbitrary natural number k. This assumption is then used to prove that the statement is also true for k+1.

Q: What is the inductive step?
A: The inductive step shows that if the statement is true for k (the inductive hypothesis), then it must also be true for k+1. This step connects the truth of the statement across all natural numbers.