A sequence is defined recursively using the formula [tex]$f(n+1)=f(n)-5$[/tex]. Which sequence could be generated using this formula?
A. [tex]$1,-5,25,-125, \ldots$[/tex]
B. [tex]$2,10,50,250, \ldots$[/tex]
C. [tex]$3,-2,-7,-12, \ldots$[/tex]
Answer (2)
Check the difference between consecutive terms in each sequence.
Sequence 1: The difference between consecutive terms is not -5.
Sequence 2: The difference between consecutive terms is not -5.
Sequence 3: The difference between consecutive terms is -5, so this sequence satisfies the recursive formula.
The sequence that could be generated using the formula f ( n + 1 ) = f ( n ) − 5 is 3 , − 2 , − 7 , − 12 , … .
Explanation
Understanding the Problem We are given a recursive formula f ( n + 1 ) = f ( n ) − 5 and three sequences. We need to determine which sequence satisfies this formula, meaning the difference between consecutive terms is always -5.
Analyzing Sequence 1 Let's examine the first sequence: 1 , − 5 , 25 , − 125 , … . We check the difference between consecutive terms:
− 5 − 1 = − 6
25 − ( − 5 ) = 30
− 125 − 25 = − 150
The difference between consecutive terms is not -5, so this sequence does not satisfy the recursive formula.
Analyzing Sequence 2 Now let's examine the second sequence: 2 , 10 , 50 , 250 , … . We check the difference between consecutive terms:
10 − 2 = 8
50 − 10 = 40
250 − 50 = 200
The difference between consecutive terms is not -5, so this sequence does not satisfy the recursive formula.
Analyzing Sequence 3 Finally, let's examine the third sequence: 3 , − 2 , − 7 , − 12 , … . We check the difference between consecutive terms:
− 2 − 3 = − 5
− 7 − ( − 2 ) = − 7 + 2 = − 5
− 12 − ( − 7 ) = − 12 + 7 = − 5
The difference between consecutive terms is -5, so this sequence satisfies the recursive formula.
Conclusion Therefore, the sequence that could be generated using the formula f ( n + 1 ) = f ( n ) − 5 is 3 , − 2 , − 7 , − 12 , … .
Examples
Recursive formulas are used in computer science to define functions that call themselves. For example, the factorial function can be defined recursively as f ( n ) = n × f ( n − 1 ) , with f ( 0 ) = 1 . This means that to calculate the factorial of a number, you multiply the number by the factorial of the number minus one, and so on, until you reach the base case of 0. Recursive formulas are also used in finance to calculate compound interest and in physics to model the motion of objects.
The sequence that satisfies the recursive formula f ( n + 1 ) = f ( n ) − 5 is 3 , − 2 , − 7 , − 12 , … because it has a consistent difference of − 5 between consecutive terms. The other two sequences do not exhibit this difference.
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