In an arithmetic progression, consecutive terms differ by a fixed constant d, so that
8 - x + d = 3x
3x + d = 4x + 1
Solving for d in both equations gives
d = 4x - 8
d = x + 1
Solve for x :
4x - 8 = x + 1
3x = 9
x = 3
Then d = 3 + 1 = 4.
So the first three terms of the sequence are 5, 9, and 13.
The nth term of the sequence is 5 + 4(n - 1) = 4n + 1, so the sum of the first 8 terms is
[tex]\displaystyle\sum_{n=1}^84n+1=4\sum_{n=1}^8n+\sum_{n=1}^81=\frac{4\cdot8\cdot9}2+8=\boxed{152}[/tex]
