Tính tổng S = s i n 2 5 0 + s i n 2 1 0 0 + s i n 2 1 5 0 + . . . + s i n 2 8 5 0

\[{\rm{S = si}}{{\rm{n}}^{\rm{2}}}{{\rm{5}}^{\rm{0}}}{\rm{ + si}}{{\rm{n}}^{\rm{2}}}{\rm{1}}{{\rm{0}}^{\rm{0}}}{\rm{ + si}}{{\rm{n}}^{\rm{2}}}{\rm{1}}{{\rm{5}}^{\rm{0}}}{\rm{ + }}...{\rm{ + si}}{{\rm{n}}^{\rm{2}}}{\rm{8}}{{\rm{5}}^{\rm{0}}}\]

\[ = (si{n^2}{5^0} + si{n^2}{85^0}) + (si{n^2}{10^0} + si{n^2}{80^0}) + ... + (si{n^2}{40^0} + si{n^2}{50^0}) + si{n^2}{45^0}\]

\[{\rm{ = }}\left[ {co{s^2}\left( {{{180}^0} - {5^0}} \right) + si{n^2}{{85}^0}} \right] + \left[ {co{s^2}\left( {{{180}^0} - {{10}^0}} \right) + si{n^2}{{80}^0}} \right] + ..\]

\[ + \left[ {co{s^2}\left( {{{180}^0} - {{40}^0}} \right) + si{n^2}{{50}^0}} \right] + si{n^2}{45^0}\]

\[{\rm{ = }}\left( {{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{8}}{{\rm{5}}^{{\rm{0 }}}}{\rm{ + si}}{{\rm{n}}^{\rm{2}}}{\rm{8}}{{\rm{5}}^{\rm{0}}}} \right){\rm{ + (co}}{{\rm{s}}^{\rm{2}}}{\rm{8}}{{\rm{0}}^{{\rm{0 }}}}{\rm{ + si}}{{\rm{n}}^{\rm{2}}}{\rm{8}}{{\rm{0}}^{\rm{0}}}{\rm{) + }}...{\rm{ + }}\left( {{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{5}}{{\rm{0}}^{{\rm{0 }}}}{\rm{ + si}}{{\rm{n}}^{\rm{2}}}{\rm{5}}{{\rm{0}}^{\rm{0}}}} \right){\rm{ + si}}{{\rm{n}}^{\rm{2}}}{\rm{4}}{{\rm{5}}^{\rm{0}}}\]

\({\rm{ = 1 + 1 + }}...{\rm{ + 1 + }}{\left( {\frac{{\sqrt {\rm{2}} }}{{\rm{2}}}} \right)^{{\rm{2 }}}}{\rm{ = 8 + }}{\left( {\frac{{\sqrt {\rm{2}} }}{{\rm{2}}}} \right)^{\rm{2}}}\)

\({\rm{ = }}\frac{{{\rm{17}}}}{{\rm{2}}}\)

Đáp án cần chọn là: B