In Class 11, the Principle of Mathematical Induction is introduced as a method of mathematical proof used to establish a given statement for all natural numbers. Here’s an overview of how it works:
1. **Statement**: The first step is to formulate the statement that you want to prove for all natural numbers. This statement is often denoted by P(n), where n is a natural number.
2. **Base Case**: The next step is to prove that the statement is true for the smallest natural number, usually 1. This is called the base case. You show that P(1) is true.
3. **Inductive Hypothesis**: Assume that the statement is true for some arbitrary natural number k. This is the inductive hypothesis: assume P(k) is true.
4. **Inductive Step**: Using the assumption that P(k) is true, you then prove that P(k+1) is also true. This step is crucial because it establishes that if the statement is true for one natural number, it must be true for the next natural number as well.
5. **Conclusion**: By the Principle of Mathematical Induction, if the statement is true for the base case (P(1)) and the inductive step (P(k) implies P(k+1)), then the statement is true for all natural numbers.
The Principle of Mathematical Induction is a powerful tool in mathematics and is used extensively in proofs involving sequences, series, divisibility, and many other areas.
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