First 7 Terms

Since $a_1, a_2, a_3, a_4$ are terms in a geometric progression, then they have a common ratio. That is, we can write $a_2 = a_1 \cdot r$ where $r$ is the common ratio.

It is given that

$a_1 + a_3 = 25$

We can write $a_3$ in terms of $a_1$ . That is, $a_3 = a_1 \cdot r^2$

As a result of substitution and factoring, we have $a_1 + a_3 = a_1 + a_1 \cdot r^2 = a_1 (1 + r^2) = 25$

The second given

$a_2 + a_4 = 50$

can also be written in the same reason.

The second given can be rewritten in the form (after substitution and factoring)

$a_2 + a_4 = a_1 \cdot r + a_1 \cdot r^3 = a_1 \cdot r \cdot(1 + r^2) = 50$

We now use the new equations

$a_1 (1 + r^2) = 25$ and $a_1 \cdot r\cdot (1 + r^2) = 50$

We take the ratio of the two equations.
$\frac{a_1\cdot r\cdot (1 + r^2)}{a_1\cdot (1 + r^2)} = \frac{50}{25}$ .

This gives us $r = 2$ . We can now compute for $a_1$ which is 5 . The geometric progression with the first seven terms consists of $\{5, 10, 20, 40, 80, 160, 320\}$ . Here we can check that $a_1 + a_3 = 5+ 20 = 25$ and $a_2 + a_3 = 10 + 40 = 50$ .

Thus the sum of these terms add up to $\boxed{635}.$