$$\frac{4^{\frac{i}{1001}}}{4^{\frac{i}{1001}}+2}+\frac{4^{\frac{1001-i}{1001}}}{4^{\frac{1001-i}{1001}}+2}$$
$$=8+2(4^{\frac{1}{1001}}+4^{\frac{1001-i}{1001}})$$
$$ดังนั้น sum = 4000+2(4^{\frac{1}{1001}}+4^\frac{2}{1001}+...+4^{\frac{1000}{1001}})=...$$
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