Tìm số hạng đầu và công sai của cấp số cộng sau, biết: \(\,\,\left\{ \begin{array}{l} {u_7} - {u_3} = 8\\ {u_2}.{u_7} = 75 \end{array} \right.\)

Ta có: u₇ = u₁ + 6d ; u₃ = u₁ + 2d ; u₂ = u₁ + d

Do đó theo đề bài ta có:

\(\begin{array}{l}\,\left\{ \begin{array}{l}
{u_7} - {u_3} = 8\\
{u_2}.{u_7} = 75
\end{array} \right. \\\Leftrightarrow \left\{ \begin{array}{l}
{u_1} + 6d - {u_1} - 2d = 8\\
\left( {{u_1} + d} \right)\left( {{u_1} + 6d} \right) = 75
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
4d = 8\\
\left( {{u_1} + d} \right)\left( {{u_1} + 6d} \right) = 75
\end{array} \right. \\ \Leftrightarrow \left\{ \begin{array}{l}
d = 2\\
\left( {{u_1} + 2} \right)\left( {{u_1} + 12} \right) = 75
\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}d = 2\\u_1^2 + 14{u_1} + 24 = 75\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}d = 2\\u_1^2 + 14{u_1} - 51 = 0\end{array} \right. \\\Leftrightarrow \left\{ \begin{array}{l}d = 2\\\left[ \begin{array}{l}{u_1} = 3\\{u_1} = - 17\end{array} \right.\end{array} \right.\end{array}\)

\( \Leftrightarrow \left[ \begin{array}{l}
\left\{ \begin{array}{l}
d = 2\\
{u_1} = 3
\end{array} \right.\\
\left\{ \begin{array}{l}
d = 2\\
{u_1} = - 17
\end{array} \right.
\end{array} \right.\)