📐 Mathematics · Principle of Mathematical Induction · JEE

While proving 2²ⁿ - 1 is divisible by 3 by induction, the inductive step rewrites 2^(2(k+1)) - 1 as:

Answer: 4(2^2k - 1) + 3.

A (2^2k - 1) + 4
B 4(2^2k - 1) + 3
C 4(2^2k - 1) - 3
D 2(2^2k - 1) + 1

Correct answer: B. 4(2^2k - 1) + 3

Explanation: 2^2k+2 - 1 = 4 . 2^2k - 1 = 4(2^2k - 1) + 3, and since 2^2k - 1 is divisible by 3 by the inductive hypothesis, the whole expression is divisible by 3.

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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While proving 22n - 1 is divisible by 3 by induction, the inductive step rewrites 2^(2(k+1)) - 1 as:

Concept context

More Principle of Mathematical Induction questions

While proving 2²ⁿ - 1 is divisible by 3 by induction, the inductive step rewrites 2^(2(k+1)) - 1 as: