lean_workbook_plus_70639

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

The characteristic polynomial of this linear recurrence relation of order 2 is $2\lambda^2 - \lambda - 1 = 0$ , of roots 1 and $-\dfrac {1} {2}$ , so the general solution is $p_n = \alpha + \beta\left ( -\dfrac {1} {2}\right )^n$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_70639  (p : ℕ → ℝ)
  (h₀ : 2 * p 2 - p 1 - 1 = 0)
  (h₁ : ∀ n, p (n + 2) = p (n + 1) - p n) :
  ∃ α β : ℝ, ∀ n, p n = α + β * (-1 / 2)^n   :=  by sorry

The characteristic equation2λ²−λ−1 was replaced by the unrelated recurrence pₙ₊₂=pₙ₊₁−pₙ.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

For2pₙ₊₂−pₙ₊₁−pₙ=0, prove pₙ=α+β(−1/2)ⁿ; determine α,β from the actual initial values. Remove the unrelated equation involving the scalar1 unless supplied by source initial data.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.