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Prompt

You are given a discrete math problem. Let F₁ = 1, F₂ = 2, and Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 3 (so 1, 2, 3, 5, 8, 13, …). For a finite set S ⊆ ℤ⁺ of indices, write Σ(S) = Σᵢ∈S Fᵢ (with Σ(∅) = 0). Call S sparse if it contains no two consecutive integers. (a) Prove by induction that Σᵢ₌₁ᵐ Fᵢ = Fₘ₊₂ − 2 for all m ≥ 1. (b) Prove by strong induction on n that every positive integer n has a sparse S with Σ(S) = n whose largest element m satisfies Fₘ ≤ n < Fₘ₊₁. (c) Prove that for every m ≥ 1, every sparse S ⊆ {1, …, m} satisfies Σ(S) < Fₘ₊₁. (d) Using (c), prove that the sparse set in (b) is unique. (e) For positive integers m, n, define m ~ n iff their sparse sets have the same largest element. Show ~ is an equivalence relation, find the sparse set for 100, and find all equivalence classes and their sizes (as a formula). (f) A student claims: “Every positive integer is Σ(S) for exactly one set S (sparseness not required), by the same argument as (d).” Show this is false, and identify exactly which step of (c) or (d) breaks. Your task is to solve the given problem by providing a clear, step-by-step solution.