Answer: the explicit rule for the geometric sequence f(1) = 9 and f(n) = (2/3) * f(n-1) for n ≥ 2 is f(n) = (2^n) / (3^(n-1)).
Step-by-step explanation:
The first term, f(1), is given as 9.
For n ≥ 2, each term f(n) is obtained by multiplying the previous term f(n-1) by 2/3.
We can express this pattern as:
f(n) = (2/3) * f(n-1)
To find the explicit rule, we need to find a general form for f(n) in terms of n. Let's write out a few terms to see if we can identify a pattern:
f(1) = 9
f(2) = (2/3) * f(1) = (2/3) * 9 = 6
f(3) = (2/3) * f(2) = (2/3) * 6 = 4
f(4) = (2/3) * f(3) = (2/3) * 4 = 8/3
f(5) = (2/3) * f(4) = (2/3) * (8/3) = 16/9
From these terms, we can see that the numerator of each term is a power of 2 (2^1, 2^2, 2^3, ...), and the denominator is a power of 3 (3^0, 3^1, 3^2, ...).
Therefore, we can write the explicit rule for the given geometric sequence as:
f(n) = (2^n) / (3^(n-1))
