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Let w_1, w_2, ..., w_6 denote the six seventh roots of unity other…
Question
Let w_1, w_2, ..., w_6 denote the six seventh roots of unity other than 1 (i.e. the non-trivial roots of z^7 = 1). Evaluate the sum 1/(3 - w_1) + 1/(3 - w_2) + ... + 1/(3 - w_6). Express the answer as a fraction m/n in lowest terms and report the value m/n.
✓ Verified answer: 1.834401checked by our engine — not a guess
Step-by-step solution
For the polynomial z^n - 1 = prod_{k=0}^{n-1}(z - w_k), logarithmic differentiation gives sum_{k=0}^{n-1} 1/(x - w_k) = n x^{n-1}/(x^n - 1).
With n = 7 and x = 3 this equals 7*3^6/(3^7 - 1) = 7*729/2186 = 5103/2186.
This sum runs over all seven roots, including w = 1, whose term is 1/(3 - 1) = 1/2.
Subtracting it: 5103/2186 - 1/2 = 5103/2186 - 1093/2186 = 4010/2186 = 2005/1093.
Since gcd(2005, 1093) = 1 the fraction is already in lowest terms.
Final answer2005/1093
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