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# Mathematical Induction

## Mathematical Induction Synopsis

Synopsis

Top Concepts

1. There are two types of reasoning — deductive and inductive.

2. In deduction, given a statement to be proven, often called a conjecture or a theorem, valid deductive steps are derived and a proof may or may not be established.

3. Deduction is the application of a general case to a particular case.

4. Inductive reasoning depends on working with each case and developing a conjecture by observing the incidence till each and every case is observed.

5. ‘Induction’ means the generalization from particular cases or facts.

6. Deductive approach is known as the ‘top-down approach’. The given theorem is narrowed down to a specific hypothesis then to observation. Finally the hypothesis is tested with specific data to get the confirmation (or not) of the original theory.

7. Inductive reasoning works the other way, moving from specific observations to broader generalizations and theories. Informally, this is known as the ‘bottom up approach’.

8. To prove statements or results formulated in terms of n, where n is a positive integer, a principle based on inductive reasoning called the Principle of Mathematical Induction is used.

9. The Principle of Mathematical Induction or PMI is one such tool which can be used to prove a wide variety of mathematical statements. Each such statement is assumed as P (n) associated with positive integer n, for which the correctness for the case n = 1 is examined. Then assuming the truth of P(k) for some positive integer k, the truth of P (k + 1) is established.

10. Let p(n) denote a mathematical statement such that
• p(1) is true i.e. the statement is true for n = 1.
• p(k + 1) is true whenever p(k) is true i.e. if the statement is true for n = k, then the statement is also true for n = k + 1.
Then the statement is true for all natural numbers ‘n’ by the Principle of Mathematical Induction.

11. PMI is based on Peano’s Axiom.

12. PMI is based on a series of well-defined steps so it is necessary to verify all of them.

13. PMI can be used to prove the equality, inequality and divisibility of natural numbers.

Key Formulae
1. Sum of n natural numbers: 1 + 2 + 3 +….+ n =

2. Sum of n2 natural numbers: 12 + 22 + 32 +………n2 =

3. Sum of odd natural numbers: 1 + 3 + 5 + 7……+ (2n - 1) = n2

4. Steps of PMI
• Denote the given statement in terms of ‘n’ by P(n).
• Check whether the proposition is true for n = 1.
• Assume that the proposition result is true for n = k.
• Using p(k), prove that the proposition is true for p(k + 1).

5. Rules of Inequalities
• If a < b and b < c then a < c.
• If a < b then a + c < b + c.
• If a < b and c > 0 which means c is positive, then ac < bc.
• If a < b and c < 0 which means c is negative, then ac > bc.