Mathematical Induction:
Let us first understand the meaning of mathematical induction,the word 'Induction' means method of reasoning from individual cases to general ones or from observed instances to unobserved ones. Many important mathematical formulae are such that a result is formed by some means which does not provide for a direct proof. Mathematical Induction is a principle by which one can arrive at a conclusion about a statement for all positive integers, after proving certain related related proposition.
Some sentences depend on a variable for its truth value (i.e., true or false).
e.g., "2+4+6+…2n=2n" is true for n=1 but false for n=2, n=3 etc.
As the above sentence is definitely true or definitely false for a particular positive integral value of n, the sentence is a statement and it depends on nÎN for its truth-value. Such statements are called predicates and are symbolised as P(n).
A statement P(n) is true for all nÎN if
(i) P(1) is true (ii) P(r) is true implies P(r+1) is true.
The following are the Illustrative Examples:
If P(n) is the statement n2-n+41 is prime, prove that P(1), P(2) are true but P(41) is not true.
Summary
1 A sentence is called a statement if it can be adjudged as true
2. Every statement is a sentence, but a sentence may or may not be a statement.
3. A statement involving natural number n is generally denoted by P(n).
Conclusion:
Let n N and P(n) denote a certain statement or formula or theorem. Then P(n) holds good for every natural number n if
(i) it holds for n = 1 and
(ii) it holds for n = k+1 whenever it holds for n = k.
Let us first understand the meaning of mathematical induction,the word 'Induction' means method of reasoning from individual cases to general ones or from observed instances to unobserved ones. Many important mathematical formulae are such that a result is formed by some means which does not provide for a direct proof. Mathematical Induction is a principle by which one can arrive at a conclusion about a statement for all positive integers, after proving certain related related proposition.
| Statements: |
e.g., "2+4+6+…2n=2n" is true for n=1 but false for n=2, n=3 etc.
As the above sentence is definitely true or definitely false for a particular positive integral value of n, the sentence is a statement and it depends on nÎN for its truth-value. Such statements are called predicates and are symbolised as P(n).
|
(i) P(1) is true (ii) P(r) is true implies P(r+1) is true.
|
If P(n) is the statement n2-n+41 is prime, prove that P(1), P(2) are true but P(41) is not true.
Summary
| or false |
3. A statement involving natural number n is generally denoted by P(n).
Conclusion:
Let n N and P(n) denote a certain statement or formula or theorem. Then P(n) holds good for every natural number n if
(i) it holds for n = 1 and
(ii) it holds for n = k+1 whenever it holds for n = k.
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