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1.Principle of Mathematical Induction
Class 11 • Mathematics • NCERT
Principle of Mathematical Induction
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This lesson introduces the Principle of Mathematical Induction as per NCERT Class 11. Students learn the idea of proving statements for all natural numbers using a logical two-step process.
Lesson Objectives
- Understand the idea of mathematical induction.
- Learn the steps of induction.
- Apply induction to prove simple results.
- Solve NCERT-style induction problems.
1. What Is Mathematical Induction?
The Principle of Mathematical Induction is a method used to prove that a statement is true for all natural numbers.
If a statement is true for n = 1 and assuming it is true for n = k implies it is true for n = k + 1, then the statement is true for all natural numbers.
2. Step 1: Base Step
In the base step, we verify that the given statement is true for the first natural number (usually n = 1).
If the statement is false for n = 1, induction cannot be applied.
3. Step 2: Inductive Hypothesis
In this step, we assume that the statement is true for some natural number n = k.
This assumption is called the inductive hypothesis.
4. Step 3: Inductive Step
Using the inductive hypothesis, we prove that the statement is true for n = k + 1.
This step completes the proof of induction.
5. Simple NCERT Example
Prove: 1 + 2 + 3 + … + n = n(n + 1)/2
• Check for n = 1
• Assume true for n = k
• Prove for n = k + 1
Practice Questions (NCERT)
- What is the Principle of Mathematical Induction?
- What is the base step?
- What is the inductive hypothesis?
- What do we prove in the inductive step?
- Is induction applicable only to natural numbers?
- Why is the base step important?
- What is assumed in induction?
- What happens if base step is false?
- Name the three main steps of induction.
- Is PMI part of NCERT Class 11 syllabus?
✅ Show Answer Key
- A method to prove statements for all natural numbers.
- Checking the statement for n = 1.
- Assuming the statement is true for n = k.
- The statement for n = k + 1.
- Yes.
- It starts the proof.
- The statement for n = k.
- Induction fails.
- Base step, hypothesis, inductive step.
- Yes.
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