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Evaluate cot( sumₙ₌₁⁴⁸ cot⁻¹(1 + sumₖ₌₁ⁿ 2k)).

1. 46/45
2. 47/48
3. 25/24
4. 23/25

Correct answer: 25/24

Solution

Each cot⁻¹(n²+n+1) equals tan⁻¹(n+1) - tan⁻¹(n), so the sum telescopes to tan⁻¹(49) - tan⁻¹(1). Taking cot of this: cot(tan⁻¹(49) - pi/4) = (49-1)/(1+49*1) = 48/50 = 24/25, and cot of the argument gives 25/24.

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