4.021. An arithmetic progression has the following property: for any $n$ the sum of its first $n$ terms is equal to $5 n^{2}$. Find the common difference of this progression and its first three terms.
## Solution.

Let $n=2$ and $n=3$. By formula (4.5) we find

$$
\left\{\begin{array} { l } 
{ \frac { 2 a _ { 1 } + d } { 2 } \cdot 2 = 5 \cdot 2 ^ { 2 } , } \\
{ \frac { 2 a _ { 1 } + 2 d } { 2 } \cdot 3 = 5 \cdot 3 ^ { 2 } }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ 2 a _ { 1 } + d = 2 0 , } \\
{ a _ { 1 } + d = 1 5 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
d=10 \\
a_{1}=5
\end{array}\right.\right.\right.
$$

Then $a_{2}=15, a_{3}=25$.

Answer: $10 ; 5,15,25$.