In an arithmetic progression a is the first term and a+2d is the third term, where d is the common difference. The eighth term is a+7d, so a+2d=2(a+7d).
a+2d=2a+14d, a+12d=0, so a=-12d. For example, if d=1 then a=-12 and the series would look like:
-12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (first 25 terms).
In general, we have a, a+d, a+2d, ..., a+22d, a+23d, a+24d for the 25 terms.
The sum of these terms is (a+a+24d)+(a+d+a+23d)+(a+2d+a+22d)+... by taking pairs of terms.
Each pair of terms adds up to 2a+24d and there are 12 pairs plus the 13th term=a+12d.
So the sum is 12(2a+24d)+a+12d=24(a+12d)+(a+12d)=25(a+12d).
We have already shown that a=-12d so the sum=25(-12d+12d)=0.
If you take another look at our example series you can see that the negatives and positives cancel out and we're left with zero as the sum.