Question:

<div class="solution"><p><strong>Step 1:</strong> \(T_k=\tan^{-1}(k+1)-\tan^{-1}k\) (telescoping via denominator = 1+k(k+1)).</p><p><strong>Step 2:</strong> Sum = \(\tan^{-1}(n+1)-\tan^{-1}1=\tan^{-1}\!\frac{n+1-1}{1+(n+1)}=\tan^{-1}\!\frac{n}{n+2}\).</p><p><strong>Answer: (A) \(\theta=n/(n+2)\)</strong></p><div class="trap-box"><strong>Trap:</strong> Don't convert entire sum at once. Telescope first.<div class="key-concept"><strong>Key Concept:</strong> Finite tan⁻^1 telescoping \to clean closed form via subtraction formula