Prove that there exists \(k ∈ ℕ\) such that, for all \(n \in ℕ\), \((n + k)^{2} ≤ 2^{n+k}\).
To do so, complete the following Lean 4 theory:
import Mathlib.Tactic example : ∃ k : ℕ, ∀ n : ℕ, (n + k) ^ 2 ≤ 2 ^ (n + k) := by sorry
Prove that, for all \(n \in ℕ\), \(2n + 9 ≤ 2^{n+4}\).
To do so, complete the following Lean 4 theory:
import Mathlib.Tactic variable (n : ℕ) example : 2 * n + 9 ≤ 2 ^ (n + 4) := by sorry
Prove in Lean 4 that 5²ⁿ - 2³ⁿ is divisible by 17 for every natural number n.
To do so, complete the following Lean 4 theory:
import Mathlib.Data.Nat.Basic import Mathlib.Tactic variable (n : ℕ) open Nat example : 17 ∣ 5^(2 * n) - 2^(3 * n) := by sorry