lean_workbook_plus_2214

⚠️ Retired — specification defect

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

Given the sequence $x_{n}=\frac{4}{7}$ if $n=0$ , and $x_{n}=7x_{(n-1)}-3$ for all positive integers n greater than zero. Find $k, p,$ and $w$ such that $x_{n}=\frac{k^{n-1}+p}{w}$ for all non-negative integers $n$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_2214 (x : ℕ → ℚ) (k p w : ℚ) (h₁ : x 0 = 4/7) (h₂ : ∀ n, n > 0 → x n = 7 * x (n - 1) - 3) (h₃ : ∀ n, x n = (k^(n - 1) + p)/w) : k = 7 ∧ p = -3 ∧ w = 7   :=  by sorry

Natural predecessor truncation makes the closed form give x₀=x₁, inconsistent with x₀=4/7 and x₁=1; the inserted parameter values are also wrong.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

For x₀=4/7 and x_(n+1)=7x_n−3 (n≥0), prove x_n=(7^n+7)/14. For the form (k^(n−1)+p)/w at n≥1 the corresponding parameters are k=7,p=1,w=2; do not impose a truncated predecessor formula at n=0.

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.