lean_workbook_plus_51099

⚠️ 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.

Find the closed form for the sequence defined by $a_{n+1} = 5a_{n} + 29b_{n}, b_{n+1} = a_{n} + 5b_{n}$, with initial conditions $a_{1} = 5, b_{1} = 1$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_51099 (a b : ℕ → ℕ) (a1 : a 0 = 5) (b1 : b 0 = 1) (a_rec : ∀ n, a (n + 1) = 5 * a n + 29 * b n) (b_rec : ∀ n, b (n + 1) = a n + 5 * b n) : ∃ A B : ℕ, ∀ n, a n = A * n + B ∧ b n = A * n + B   :=  by sorry

Use a_n+√29*b_n=(5+√29)^n when the initial terms are indexed by1. If the formal sequences start at0 with the same initial values, use exponent n+1; obtain a_n,b_n from the two conjugate powers rather than affine formulas.

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

Proposed corrected statement

Use a_n+√29*b_n=(5+√29)^n when the initial terms are indexed by1. If the formal sequences start at0 with the same initial values, use exponent n+1; obtain a_n,b_n from the two conjugate powers rather than affine formulas.

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.