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Principles of Mathematical Induction: Mathematical induction is a process of reasoning that allows us to establish a result for a given sequence of numbers. This result is then used to prove a statement about the sequence.
Explain in Detail: Properties of Mathematical Induction
The basis of mathematical induction is the fact that if a statement is true for the first number in a sequence, it is also true for the second number. This is because the second number is just a copy of the first number, with one addition or subtraction.
To use mathematical induction, we first need to come up with a statement that we want to prove. This statement can be about a sequence of numbers, or about a property that the numbers in the sequence have.
We then need to come up with a base case, which is a number for which we know the statement is definitely true. We can then use mathematical induction to prove the statement is true for all other numbers in the sequence.
Principles of Mathematical Induction Examples
The principle of mathematical induction is a logical principle that states that a statement is true for all natural numbers if it is true for the first natural number and for every natural number that follows the first natural number.
Some examples of how the principle of mathematical induction can be used are as follows:
– To show that a statement is true for all natural numbers, it is first necessary to show that the statement is true for the first natural number.
– Then, it is necessary to show that the statement is true for every natural number that follows the first natural number.
– If both of these steps can be shown to be true, then the statement is said to be true for all natural numbers.
The principle of mathematical induction is a logical principle that is used to prove a statement is true for all natural numbers. The principle is based on the assumption that if a statement is true for a particular natural number, then it is also true for the next natural number. The principle can be used to prove a statement is true for all positive integers.
Also Read: Principle Of Mathematical Induction Questions for CBSE Class 11th
The principle of mathematical induction can be illustrated with the following example.
Example:
Prove that the statement “2 + 3 = 5” is true for all natural numbers.
The statement “2 + 3 = 5” is true for the natural number 2. The principle of induction can be used to show that the statement is also true for the next natural number, which is 3.
