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Let w = e^{2*pi*i/10} be a primitive 10th root of unity. Evaluate the…
Question
Let w = e^{2*pi*i/10} be a primitive 10th root of unity. Evaluate the sum S = (1 + w^0)^25 + (1 + w^1)^25 + (1 + w^2)^25 + ... + (1 + w^9)^25, the sum taken over all ten 10th roots of unity. Report the exact integer value of S.
✓ Verified answer: 33218910checked by our engine — not a guess
Step-by-step solution
Expand each term by the binomial theorem: (1 + w^k)^25 = sum_{j=0}^{25} C(25, j) w^{kj}.
Summing over the ten 10th roots and swapping order: S = sum_{j=0}^{25} C(25, j) (sum_{k=0}^{9} w^{kj}).
The inner geometric sum over all 10th roots equals 10 when 10 | j and 0 otherwise.
The multiples of 10 in [0, 25] are j = 0, 10, 20.
Hence S = 10*(C(25,0) + C(25,10) + C(25,20)) = 10*(1 + 3268760 + 53130) = 10*3321891 = 33218910.
Final answer33218910
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