4.007. When the ninth term of an arithmetic progression is divided by the second term, the quotient is 5, and when the thirteenth term is divided by the sixth term, the quotient is 2 and the remainder is 5. Find the first term and the common difference of the progression.
Solution.

From the condition we have $\left\{\begin{array}{l}a_{9}=5 a_{2}, \\ a_{13}=2 a_{6}+5 .\end{array}\right.$

Using formula (4.1), we get

$$
\left\{\begin{array} { l } 
{ a _ { 1 } + 8 d = 5 ( a _ { 1 } + d ) , } \\
{ a _ { 1 } + 1 2 d = 2 ( a _ { 1 } + 5 d ) + 5 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
4 a_{1}=3 d, \\
a_{1}=2 d-5,
\end{array} \Leftrightarrow\right.\right.
$$

$$
\left\{\begin{array} { l } 
{ 4 ( 2 d - 5 ) = 3 d , } \\
{ a _ { 1 } = 2 d - 5 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
d=4 \\
a_{1}=3
\end{array}\right.\right.
$$

Answer: $3 ; 4$.