What is Mathematical Induction?
Induction means reasoning from particular cases (observations) to a general statement.
A statement may look true from a few examples, but to prove it is true for all natural numbers, we need Mathematical Induction.
If it's false, we can disprove it by finding a counter-example.
A statement may look true from a few examples, but to prove it is true for all natural numbers, we need Mathematical Induction.
If it's false, we can disprove it by finding a counter-example.
Proposition P(n)
A statement whose truth value depends on a natural number n is denoted as P(n).
Example:
Example:
1² + 2² + 3² + ... + n² = n(n+1)(2n+1)⁄6
We write this as P(n), and P(4) means checking for n = 4.
First Principle of Mathematical Induction
To prove P(n) is true for all natural numbers n ≥ k, we must show:
1
Base Case (Verification Step)
P(k) is true
(Usually k = 1, but can be any starting natural number)
2
Induction Hypothesis
Assume P(m) is true for some m ≥ k
3
Induction Step
Prove that P(m+1) is also true
Conclusion: By the Principle of Mathematical Induction,
P(n) is true for all natural numbers n ≥ k
P(n) is true for all natural numbers n ≥ k
Standard Procedure to Apply Induction
- Step 1 (Base Case): Verify P(1) is true (or P(k) for starting value k)
- Step 2 (Assumption): Assume P(m) is true for some integer m ≥ 1 (or m ≥ k)
- Step 3 (Induction Step): Using the assumption, prove that P(m+1) is true
- Conclusion: Hence, by mathematical induction, P(n) is true for all natural numbers n ≥ 1 (or n ≥ k)
Common Mistakes to Avoid
- Forgetting to check the base case
- Assuming P(1) is true without verifying
- Proving P(m+2) instead of P(m+1)
- Using strong induction when only weak (first principle) is needed
- Wrongly concluding for all n when base case fails
JEE Main Key Points
Most Frequently Tested:
• Sum of series (1² + 2² + ... + n²), (1 + 2 + ... + n)
• Divisibility problems (like 32n – 1 is divisible by 8)
• Inequalities (AM–GM, Bernoulli, etc.)
• Matrix or determinant properties (rare but high weightage)
• Sum of series (1² + 2² + ... + n²), (1 + 2 + ... + n)
• Divisibility problems (like 32n – 1 is divisible by 8)
• Inequalities (AM–GM, Bernoulli, etc.)
• Matrix or determinant properties (rare but high weightage)
- Always write: "Let P(n) be the statement..."
- Clearly mention Base Case, Assumption, and Proof for n = m+1
- Conclusion must include "By principle of mathematical induction"
- If starting from n = 0 or n = 2, mention it clearly