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,-1.2,0.6, \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 .
Identify the recursive relationship: f ( n + 1 ) = − 0.5 f ( n ) .
Match the derived formula with the given options.
The correct recursive formula is: f ( n + 1 ) = ( − 0.5 ) f ( n ) .
Explanation
Understanding the Problem We are given a sequence 9.6 , − 4.8 , 2.4 , − 1.2 , 0.6 , … and we want to find a recursive formula that generates this sequence, given that f ( 1 ) = 9.6 and n ≥ 1 . A recursive formula defines a term in the sequence based on the previous term(s).
Finding the Pattern To find the recursive formula, we need to determine the relationship between consecutive terms. Let's 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
Since the ratio between consecutive terms is constant and equal to − 0.5 , this suggests 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 .
Matching the Formula Now, let's compare our derived formula with the given options:
f ( n + 1 ) = ( − 0.5 ) f ( n )
f ( n + 1 ) = ( 0.5 ) f ( n )
f ( n + 1 ) = f ( 0.5 n )
f ( n + 1 ) = f ( − 0.5 n )
The first option, f ( n + 1 ) = ( − 0.5 ) f ( n ) , matches our derived formula.
Examples
Recursive formulas are used in computer science to define functions that call themselves, breaking down complex problems into simpler, self-similar subproblems. For example, calculating the factorial of a number or traversing a tree structure can be elegantly implemented using recursion. In finance, recursive formulas can model compound interest or the depreciation of assets over time.
The recursive formula that generates the sequence 9.6 , − 4.8 , 2.4 , − 1.2 , 0.6 , … is f ( n + 1 ) = ( − 0.5 ) f ( n ) . This means each term in the sequence is obtained by multiplying the previous term by − 0.5 . Therefore, the correct choice is option A.
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