There are two ways of reasoning that is commanly used in drawing scientific conclusions.
• One is induction and another is deduction.
• Induction is a logical reasoning process used to establish a general proposition or statement based on specific cases or instances.
• Deduction, also known as deductive reasoning, is a method of logical inference where specific conclusions are drawn from general premises.
→ Types of Induction:
• Mathematical Induction
• Strong Induction
→ Mathematical Induction:
Mathematical Induction is an extremely important proof technique that is used to prove the mathematical theorems or statements.
• In many branches of mathematics such as algebra, geometry, arithmetic, etc , the statements or theorems involving positive integers which cannot be derived by direct proof can be proved by applying the method of mathematical induction.
• Mathematical induction typically involves three main steps:
• Base Case:
Proving that the statement is true for a specific initial value (usually the smallest value of the set of natural numbers). This step establishes that the statement holds for at least one instance.
In this step, we need to verify that P(n) is true for n=0 or 1.
• Induction Hyptothesis:
In this step, we assume that P(n) is ture for n=k i.e. P(k) is true.
• Inductive Step:
Assuming that the statement is true for a particular value (usually an arbitrary value k), and then proving that it’s true for the next value(K+1).
In this step, by using induction hypothesis, we show that P(n) is true for n=K+1.
If the base case, induction hypothesis and the inductive step are successfully established, it can be concluded that the statement is true for all natural numbers starting from the chosen initial value. This follows from the principle that if the statement is true for k, and if it can be shown that its truth for k+1 follows from its truth for k, then it must be true for all values beyond k.
In summary,
Denote the given statement(proposition) by P(n).
Prove that P(1) is true. → <strong>Basic Step</strong>
Assume that P(k) is true, Just replace n by k in P(n). → <strong>Induction Hypothesis</strong>
Show that P(k+1) is also true. → <strong>Inductive Step</strong>Then by the principle of mathematical induction P(n) is true for all n∈N, where n is a natural number.
Example:
