tim tong :
S=[TEX]C_(2009)^0+C_(2009)^4+C_(2009)^8+......................+C_(2009)^(2004)+C_(2009)^(2008)[/TEX]
[TEX]\begin{array}{l}A = C_{2009}^1 + C_{2009}^5 + .. + C_{2009}^{2009}\\B = C_{2009}^2 + C_{2009}^6 + .. + C_{2009}^{2006}\\C = C_{2009}^3 + C_{2009}^7 + .. + C_{2009}^{2007}\end{array}[/TEX]
[TEX]\begin{array}{l}{\left( {1 + 1} \right)^{2009}} = C_{2009}^0 + C_{2009}^1 + C_{2009}^2 + C_{2009}^3 + C_{2009}^4 + ... + C_{2009}^{2009} = VT + A + B + C\\{\left( {1 + i} \right)^{2009}} = C_{2009}^0 + C_{2009}^1i + C_{2009}^2{i^2} + C_{2009}^3{i^3} + C_{2009}^4{i^4} + ... + C_{2009}^{2009}{i^{2009}} = VT + iA + {i^2}B + {i^3}C\\{\left( {1 + {i^2}} \right)^{2009}} = C_{2009}^0 + C_{2009}^1\left( {{i^2}} \right) + C_{2009}^2{\left( {{i^2}} \right)^2} + C_{2009}^3{\left( {{i^2}} \right)^3} + C_{2009}^4{\left( {{i^2}} \right)^4} + ... + C_{2009}^{2009}{\left( {{i^2}} \right)^{2009}} = VT + {i^2}A + B + {i^2}C\\{\left( {1 + {i^3}} \right)^{2009}} = C_{2009}^0 + C_{2009}^1\left( {{i^3}} \right) + C_{2009}^2{\left( {{i^3}} \right)^2} + C_{2009}^3{\left( {{i^3}} \right)^3} + C_{2009}^4{\left( {{i^3}} \right)^4} + ... + C_{2009}^{2009}{\left( {{i^3}} \right)^{2009}} = VT + {i^3}A + {i^2}B + iC\\ = > {\left( {1 + i} \right)^{2009}} + {\left( {1 + {i^2}} \right)^{2009}} + {\left( {1 + {i^3}} \right)^{2009}} + {2^{2009}} = 4VT + \left( {1 + i + {i^2} + {i^3}} \right)\left( {A + C} \right) + 2\left( {1 + {i^2}} \right)C = 4VT\\ = > VT = \frac{{{{\left( {1 + i} \right)}^{2009}} + {2^{2009}} + {{\left( {1 - i} \right)}^{2009}}}}{4} = \frac{{{2^{2009}} + {{\left[ {\sqrt 2 \left( {c{\rm{os}}\frac{\pi }{4} + i\sin \frac{\pi }{4}} \right)} \right]}^{2009}} + {{\left[ {\sqrt 2 \left( {c{\rm{os}}\frac{\pi }{4} - i\sin \frac{\pi }{4}} \right)} \right]}^{2009}}}}{4} = {2^{2007}} + {2^{1002}}\end{array}[/TEX]
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