Cho cấp số cộng (u_n) biết: \(\left\{ {\begin{array}{*{20}{c}}{{{\rm{u}}_{\rm{7}}} - {{\rm{u}}_{\rm{3}}}{\rm{ = 8}}}\\{{{\rm{u}}_{\rm{2}}}{{\rm{u}}_{\rm{7}}}{\rm{ = 75}}}\end{array}} \right.\). Chọn đáp án đúng.

\(\left\{ {\begin{array}{*{20}{c}}{{{\rm{u}}_{\rm{7}}} - {{\rm{u}}_{\rm{3}}}{\rm{ = 8}}}\\{{{\rm{u}}_{\rm{2}}}{{\rm{u}}_{\rm{7}}}{\rm{ = 75}}}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{\left( {{{\rm{u}}_{\rm{1}}}{\rm{ + 6d}}} \right) - \left( {{{\rm{u}}_{\rm{1}}}{\rm{ + 2d}}} \right){\rm{ = 8}}}\\{\left( {{{\rm{u}}_{\rm{1}}}{\rm{ + d}}} \right)\left( {{{\rm{u}}_{\rm{1}}}{\rm{ + 6d}}} \right){\rm{ = 75}}}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{{\rm{4d = 8}}}\\{\left( {{{\rm{u}}_{\rm{1}}}{\rm{ + d}}} \right)\left( {{{\rm{u}}_{\rm{1}}}{\rm{ + 6d}}} \right){\rm{ = 75}}}\end{array}} \right.\)

\( \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{{\rm{d = 2}}}\\{\left( {{{\rm{u}}_{\rm{1}}}{\rm{ + d}}} \right)\left( {{{\rm{u}}_{\rm{1}}}{\rm{ + 6d}}} \right){\rm{ = 75(*)}}}\end{array}} \right.\)

Thế d = 2 vào (*) ta được:

\[\left( {{{\rm{u}}_{\rm{1}}}{\rm{ + 2}}} \right)\left( {{{\rm{u}}_{\rm{1}}}{\rm{ + 6}}{\rm{.2}}} \right){\rm{ = 75}} \Leftrightarrow \left( {{{\rm{u}}_{\rm{1}}}{\rm{ + 2}}} \right)\left( {{{\rm{u}}_{\rm{1}}}{\rm{ + 12}}} \right){\rm{ = 75}} \Leftrightarrow {\rm{u}}_1^2{\rm{ + 2}}{{\rm{u}}_{\rm{1}}}{\rm{ + 12}}{{\rm{u}}_{\rm{1}}}{\rm{ + 24 = 75}}\]

\[ \Leftrightarrow {\rm{u}}_1^2{\rm{ + 14}}{{\rm{u}}_{\rm{1}}} - {\rm{51 = 0}} \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{{{\rm{u}}_{{\rm{1 }}}}{\rm{ = }} - 1}\\{{{\rm{u}}_{\rm{1}}}{\rm{ = 3}}}\end{array}} \right.\]

Đáp án cần chọn là: C