What is the formula for the sum of a geometric series?
Not using any memorized formulas, just thinking about it:
Given that 6r^n = 4374, divide through by 6 to get r^n = 729. Note that 729 = 27^2, or 9^3, or 3^6; so the answer must be that either r=27 and n=2, r=9 and n=3, or r=3 and n=6. See which one results in a series whose sum of terms 6 + 6r + 6r^2 + ...6r^n = 6558.