Proving 2ⁿ > n² for n ≥ 5

Intro to Proofs · Axiom Academy

Use mathematical induction to prove an exponential inequality Prove that 2^n > n^2 for every integer . Exponentials eventually overtake polynomials — induction lets us pin down that the crossover holds from n = 5 onward and never reverses. Nice work — you built a complete induction proof for an inequality. Here is what made it go through: Base Case: Verify the smallest value in the domain. Here we checked n = 5 and confirmed 2^5 = 32 > 25 = 5^2 . Inductive Hypothesis: Assume the claim for some arbitrary , giving us 2^k > k^2 to work with. Algebraic Strategy: Write , then apply the hypothesis to get . Auxiliary Inequality: The key was showing 2k^2 > (k+1)^2 for , which simplifies to k^2 - 2k - 1 > 0 . Why : The claim fails at n = 2, 3, 4 (where ), so n = 5 is exactly where the exponential pulls ahead for good. This pattern shows up whenever you compare exponential and polynomial growth: the exponential eventually dominates — induction nails down precisely where the crossover happens and proves it never reverses.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.