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Let S = Σ (from k = 1 to 20) of arctan(1/(k² + k + 1)). The value of…
Question
Let S = Σ (from k = 1 to 20) of arctan(1/(k² + k + 1)). The value of tan S can be written as m/n in lowest terms with m, n positive integers. Find m + n.
✓ Verified answer: 21checked by our engine — not a guess
Step-by-step solution
Note that 1/(k² + k + 1) = ((k+1) − k)/(1 + k(k+1)), which is exactly the tangent-subtraction form: arctan(1/(k²+k+1)) = arctan(k+1) − arctan(k).
So S telescopes: S = Σ (k=1..20)[arctan(k+1) − arctan(k)] = arctan(21) − arctan(1).
Then tan S = tan(arctan 21 − arctan 1) = (21 − 1)/(1 + 21·1) = 20/22 = 10/11.
In lowest terms m/n = 10/11, so m + n = 10 + 11 = 21.
Final answer21
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