For[i = 1, i <= 5, i++,
{sol =
DSolve[{Subscript[c'[t],
i] == ((p*n*F)/v)*(Subscript[c[t], i - 1] -
Subscript[c[t], i]), Subscript[c[t], 0] == 0,
Subscript[c[0], 1] == (p*n*Q)/v,
Subscript[c[0], i + 1] == 0}, Subscript[c[t], i], t]}]

我随便看了一眼，建议是
1. 应写作 Subscript[c, i][t]
2. 不使用下标