Define the terms for an arithmetic progression.

Let $a$ be the first term and $d$ be the common difference.
The $n$-th term is given by $T_n = a + (n-1)d$.
The sum of the first $n$ terms is given by $S_n = \frac{n}{2}(2a + (n-1)d)$.

From the statement "The eleventh term of an arithmetic progression is four times its second term":

From the statement "The sum of the first seven terms of the same progression is 175":

Multiply both sides by $\frac{2}{7}$:

Divide by 2:

Substitute $a = 2d$ into $a + 3d = 25$:

The first term is 10 and the common difference is 5.

The final answer is $\text{First term} = 10, \text{Common difference} = 5$.