Which recursive formula can be used to generate the sequence shown, where [tex]f(1)=9.6[/tex] and [tex]n \geq 1[/tex]? [tex]9.6,-4.8,2.4,-12,0.6, \ldots[/tex]
A. [tex]f(n+1)=(-0.5) f(n)[/tex]
B. [tex]f(n+1)=(0.5) f(n)[/tex]
C. [tex]f(n+1)=f(0.5 n)[/tex]
D. [tex]f(n+1)=f(-0.5 n)[/tex]
Answer (2)
Calculate the ratio between consecutive terms: 9.6 − 4.8 = − 0.5 , − 4.8 2.4 = − 0.5 , 2.4 − 1.2 = − 0.5 , − 1.2 0.6 = − 0.5 .
Observe that the ratio between consecutive terms is constant and equal to − 0.5 .
Write the recursive formula as f ( n + 1 ) = − 0.5 ( n ) .
The recursive formula is f ( n + 1 ) = ( − 0.5 ) f ( n ) .
Explanation
Understanding the Sequence We are given a sequence 9.6 , − 4.8 , 2.4 , − 1.2 , 0.6 , … with f ( 1 ) = 9.6 , and we need to find the recursive formula that generates this sequence. A recursive formula defines a term based on the previous term(s).
Finding the Ratio To find the recursive formula, let's examine the ratio between consecutive terms:
Ratio between the first two terms: 9.6 − 4.8 = − 0.5 Ratio between the second and third terms: − 4.8 2.4 = − 0.5 Ratio between the third and fourth terms: 2.4 − 1.2 = − 0.5 Ratio between the fourth and fifth terms: − 1.2 0.6 = − 0.5
Since the ratio between consecutive terms is constant and equal to − 0.5 , we can conclude that each term is obtained by multiplying the previous term by − 0.5 .
Recursive Formula Therefore, the recursive formula is f ( n + 1 ) = − 0.5 ⋅ f ( n ) . This means that to find the next term in the sequence, we multiply the current term by − 0.5 .
Final Answer The recursive formula that generates the given sequence is f ( n + 1 ) = ( − 0.5 ) f ( n ) .
Examples
Recursive formulas are used in computer science to define functions that call themselves, breaking down a problem into smaller, self-similar subproblems. For example, calculating the factorial of a number can be defined recursively: f a c t or ia l ( n ) = n ∗ f a c t or ia l ( n − 1 ) , with a base case of f a c t or ia l ( 1 ) = 1 . This concept is also used in financial modeling to predict future values based on past performance and a growth rate, or in population dynamics to model population growth or decline.
The recursive formula that generates the sequence is f ( n + 1 ) = ( − 0.5 ) f ( n ) , indicating that each term is obtained by multiplying the previous term by − 0.5 . Therefore, the answer is option A.
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