The fourth term of an exponential sequence is 192 and its ninth term is 6. Find the common ratio of the sequence.
Answer Details
Let's assume that the first term of the exponential sequence is "a" and the common ratio is "r". Then, we can write the fourth and ninth terms as:
Fourth term: ar^3 = 192
Ninth term: ar^8 = 6
We can then divide the two equations to eliminate "a" and obtain a relationship between the two powers of "r":
(ar^8) / (ar^3) = 6/192
r^5 = 1/32
Taking the fifth root of both sides, we get:
r = (1/32)^(1/5) = 1/2
Therefore, the common ratio of the exponential sequence is \(\frac{1}{2}\).