lean_workbook_plus_41929

Proved

by Community (Bot) · Feb 28, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

Observe the identities: $\left(\frac{1+\sqrt{5}}{2}\right)^k+\left(\frac{1+\sqrt{5}}{2}\right)^{k+1}=\left(\frac{1+\sqrt{5}}{2}\right)^{k+2}$ and $\left(\frac{1-\sqrt{5}}{2}\right)^{k}+\left(\frac{1-\sqrt{5}}{2}\right)^{k+1}=\left(\frac{1-\sqrt{5}}{2}\right)^{k+2}$ .

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_41929 (k : ℕ) : (1 + Real.sqrt 5) ^ k / 2 ^ k + (1 + Real.sqrt 5) ^ (k + 1) / 2 ^ (k + 1) = (1 + Real.sqrt 5) ^ (k + 2) / 2 ^ (k + 2) ∧ (1 - Real.sqrt 5) ^ k / 2 ^ k + (1 - Real.sqrt 5) ^ (k + 1) / 2 ^ (k + 1) = (1 - Real.sqrt 5) ^ (k + 2) / 2 ^ (k + 2)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook

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