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📐 Mathematics  ·  Principle of Mathematical Induction  ·  JEE

For proving 2<sup>n</sup> > n for all natural numbers n, what is P(1)?

Answer: 2 > 1.

  • A 2 > 1
  • B 1 > 2
  • C 2 = 1
  • D 1 < 0

Correct answer: A. 2 > 1

Explanation: Substituting n=1: 2<sup>1</sup> = 2, and 2 > 1 is true, so the base case holds.

Induction as a Domino ChainP(1): base case fallsP(k) knocks down P(k+1)...and so on, foreverBase case = first domino tipped; inductive step = each domino guaranteed to tip the next one

Mathematical induction works like a row of dominoes: proving the base case P(1) tips the first domino, and proving the inductive step (P(k) ⟹ P(k+1)) guarantees each domino knocks over the next - together these two facts guarantee ALL dominoes fall, without checking each one individually.

Concept context

A proof technique used to establish that a statement is true for every natural number, using a base case and an inductive step.

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