Answer:
27(2/3)ⁿ
Explanation:
Here is the complete question
The formula f(x + 1) = 2/3(f(x)) defines a geometric sequence where f(1) = 18. which explicit formula can be used to model the same sequence?
Solution
The general formula for a geometric sequence is U = arⁿ⁻¹
From f(x + 1) = 2/3(f(x)),
f(2) = 2/3(f(1)) = ar = second term and f(1) = 18 = a = first term, r = common ratio
r = ar/a = f(2)/f(1) = 2/3
U = arⁿ⁻¹ = 18(2/3)ⁿ⁻¹
= 2 × 3² × (2/3)ⁿ⁻¹
= 2 × 3² × 2ⁿ⁻¹ × (1/3)ⁿ⁻¹
= 2 × 2ⁿ⁻¹ × 3² ×(1/3)ⁿ⁻¹
= 2ⁿ × 3³/3ⁿ
= 3³(2/3)ⁿ
= 27(2/3)ⁿ