lean_workbook_plus_43785

⚠️ 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.

Sum the series\n\n$\frac{1}{4}+\frac{1}{4}+\frac{9}{64}+\frac{1}{16}+\frac{25}{1024}+\ldots$\n\nProposed by droid347\n\nI'm a bit stumped here. But we can write this as:\n\n$\frac{1^2}{4^1}+\frac{2^2}{4^2}+\frac{3^2}{4^3}+\frac{4^2}{4^4}+\frac{5^2}{4^5}...$\n\nif you're familiar with arithmetico-geometric sequences, this can be solved in a similar fashion.\n\nNote that quadratic functions have a second common difference.\n\nIf we take $S=\frac{1^2}{4^1}+\frac{2^2}{4^2}+\frac{3^2}{4^2}+...$ we can multiply both sides by $\frac{1}{4}$ and subtract $\frac{1}{4}S$ from $S$ to get a sequence with linearly increasing numerators and common ratio: $\frac{3}{4}S=\frac{1}{4}+\frac{3}{16}+\frac{5}{64}+...$ . We can multiply $\frac{3}{4}S$ by $\frac{1}{4}$ again and subtract from $\frac{3}{4}S$ to finally get a sequence that looks like: $\frac{9}{16}S=\frac{1}{4}+\frac{2}{16}+\frac{2}{64}+\frac{2}{256}+...$ and this is clearly a single term plus an infinite geometric sequence, so the rest of the computation is easy.\n\nWe can find that as\n\n$\frac{9}{16}S=1/4+1/6$ after you sum the infinite geometric sequence. Solving for $S$ gives\n\n$S=\frac{20}{27}$ , so our answer is $\boxed{(C)}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_43785 :
  ∑' k : ℕ, (↑(k + 1))^2 / (4^(k + 1)) = 20 / 27   :=  by sorry

The ambiguous cast still elaborates to natural arithmetic and proves0=0.

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

Explicitly type the squared numerator and division in ℝ before summation; retain the correct real sum20/27.

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.