Question:

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