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Principle of Mathematical Induction

NIOS · Class 12 · Mathematics

Quick revision notes for Principle of Mathematical Induction — NIOS Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.

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Key Topics to Revise

1

1. Understanding Statements and Propositions

  • A statement (or proposition) is a sentence that is either definitively true or definitively false, but never both
  • Not all sentences are statements - questions, exclamations, and open sentences without fixed domains are NOT statements
  • Open sentences like 'x - 5 = 7' become statements only when the variable's domain is specified or when quantified
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2. The Principle of Mathematical Induction - Statement and Understanding

  • Mathematical induction consists of TWO MUST-SATISFY conditions that work together like climbing a ladder
  • BASE CASE: P(1) must be true - this is the first rung of the ladder
  • INDUCTION CASE: IF P(k) is true, THEN P(k+1) must be true - this is the mechanism that takes you from one rung to the next
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3. Proofs Using Mathematical Induction - Sum Formulas

  • SUM FORMULA PROBLEMS: Prove that the sum of certain sequences equals a given formula
  • Common patterns: arithmetic sequences (1+2+...+n), odd numbers (1+3+5+...), cubes (1³+2³+3³+...)
  • Strategy: Use induction hypothesis to substitute the known formula for sum up to k, then show sum to k+1 follows the pattern
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4. Proofs Using Mathematical Induction - Divisibility

  • DIVISIBILITY PROBLEMS: Prove that some expression is divisible by a given number for all natural numbers n
  • A number is divisible by d if we can write it as d×m for some integer m
  • Strategy: Assume P(k) says the expression with k is divisible by d (equals d×m for some integer m), then show P(k+1) is also divisible by d

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Full Notes

Key Concepts

A mathematical statement (or proposition)The Principle of Mathematical Induction statesThe base case is the firstThe inductive step involves two partsEvery proof by mathematical induction follows

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Sources & Official References

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