Airforce X Y / Indian Navy SSR Exam > Airforce X Y / Indian Navy SSR Videos > Mathematics for Airmen Group X > Examples: Principle of Mathematical Induction - 1
Examples: Principle of Mathematical Induction - 1 Video Lecture | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR
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| 1. What is the Principle of Mathematical Induction? |  |
Ans. The Principle of Mathematical Induction is a mathematical proof technique used to prove statements about natural numbers. It consists of two steps: the base step and the inductive step. In the base step, we show that the statement holds true for the initial value, usually 0 or 1. In the inductive step, we assume that the statement holds true for some arbitrary value, and then prove that it also holds true for the next value. By repeating this process, we can prove that the statement is true for all natural numbers.
| 2. How is the Principle of Mathematical Induction used in the JEE exam? | |
Ans. The Principle of Mathematical Induction is a fundamental concept in algebra and number theory, which are important topics in the JEE exam. Questions related to sequences, series, divisibility, inequalities, and other mathematical concepts often require the use of mathematical induction to prove the given statements. Understanding and applying the Principle of Mathematical Induction is crucial for solving these types of problems in the JEE exam.
| 3. Can you provide an example of using the Principle of Mathematical Induction in the JEE exam? | |
Ans. Sure! Here's an example: Prove that for any positive integer n, 7^n - 1 is divisible by 6.
Base Step: For n = 1, 7^1 - 1 = 6, which is divisible by 6.
Inductive Step: Assume that for some arbitrary positive integer k, 7^k - 1 is divisible by 6.
We need to prove that 7^(k+1) - 1 is divisible by 6.
Using the assumption, we can write 7^(k+1) - 1 = 7^k * 7 - 1 = (6 + 1)^k * 7 - 1 = (6m + 1) * 7 - 1 = 42m + 7 - 1 = 42m + 6.
Since 42m + 6 is divisible by 6, we have proved that 7^(k+1) - 1 is divisible by 6.
Therefore, by the Principle of Mathematical Induction, we can conclude that for any positive integer n, 7^n - 1 is divisible by 6.
| 4. Are there any limitations to using the Principle of Mathematical Induction? | |
Ans. Yes, there are some limitations to using the Principle of Mathematical Induction. It can only be used to prove statements about natural numbers. It cannot be used to prove statements about real numbers, irrational numbers, or negative numbers. Additionally, the Principle of Mathematical Induction can only be used when the base step is true and the inductive step can be proven. If either of these conditions is not met, the Principle of Mathematical Induction cannot be used to prove the statement.
| 5. Can the Principle of Mathematical Induction be used to prove all mathematical statements? | |
Ans. No, the Principle of Mathematical Induction cannot be used to prove all mathematical statements. It is a specific proof technique that is applicable only to statements that can be expressed in terms of natural numbers. There are many other proof techniques, such as direct proof, proof by contradiction, and proof by contrapositive, that are used to prove statements in different areas of mathematics. The choice of proof technique depends on the nature of the statement and the mathematical context in which it is being studied.
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