A sequence is defined by the recursive function [tex]f(n+1)=\frac{1}{3} f(n)[/tex]. If [tex]f(3)=9[/tex], what is [tex]f(1)[/tex]?
Answer (2)
Use the recursive formula f ( n + 1 ) = 3 1 f ( n ) and the given value f ( 3 ) = 9 .
Find f ( 2 ) by using f ( 3 ) = 3 1 f ( 2 ) , which gives f ( 2 ) = 3 × f ( 3 ) = 3 × 9 = 27 .
Find f ( 1 ) by using f ( 2 ) = 3 1 f ( 1 ) , which gives f ( 1 ) = 3 × f ( 2 ) = 3 × 27 = 81 .
The value of f ( 1 ) is 81 .
Explanation
Understanding the Problem We are given a recursive sequence defined by f ( n + 1 ) = 3 1 f ( n ) and f ( 3 ) = 9 . Our goal is to find the value of f ( 1 ) .
Finding f(2) To find f ( 1 ) , we need to work backwards using the recursive formula. First, we find f ( 2 ) using the given value of f ( 3 ) . Since f ( n + 1 ) = 3 1 f ( n ) , we can write f ( 3 ) = 3 1 f ( 2 ) .
Calculating f(2) We know that f ( 3 ) = 9 , so we can substitute this into the equation: 9 = 3 1 f ( 2 ) . To solve for f ( 2 ) , we multiply both sides of the equation by 3: 3 × 9 = 3 × 3 1 f ( 2 ) 27 = f ( 2 ) Thus, f ( 2 ) = 27 .
Finding f(1) Now that we have f ( 2 ) , we can find f ( 1 ) using the recursive formula again. We have f ( 2 ) = 3 1 f ( 1 ) .
Calculating f(1) We know that f ( 2 ) = 27 , so we can substitute this into the equation: 27 = 3 1 f ( 1 ) . To solve for f ( 1 ) , we multiply both sides of the equation by 3: 3 × 27 = 3 × 3 1 f ( 1 ) 81 = f ( 1 ) Thus, f ( 1 ) = 81 .
Final Answer Therefore, the value of f ( 1 ) is 81.
Examples
Recursive sequences are used in many areas of mathematics and computer science. For example, the Fibonacci sequence (1, 1, 2, 3, 5, 8, ...) is a recursive sequence where each term is the sum of the two preceding terms. In finance, compound interest can be modeled using a recursive sequence where the balance at the end of each period is a function of the balance at the end of the previous period. Understanding recursive sequences helps in modeling and predicting patterns in various real-world phenomena.
To find f ( 1 ) using the recursive formula f ( n + 1 ) = 3 1 f ( n ) with f ( 3 ) = 9 , we first determine f ( 2 ) as 27. We then calculate f ( 1 ) as 81. Thus, f ( 1 ) = 81 .
;
