A sequence is defined by the recursive function [tex]f(n+1)=f(r)-2[/tex]. If [tex]f(t)=10[/tex], what is [tex]f(3)[/tex]?

The problem gives a recursive function f ( n + 1 ) = f ( n ) − 2 and f ( t ) = 10 .
Express f ( 3 ) in terms of f ( t ) as f ( 3 ) = f ( t ) − 2 ( 3 − t ) .
Test the answer choices to see which one is possible for some value of t .
Assuming t = 2 , we find that f ( 3 ) = 8 , so the final answer is 8 ​ .

Explanation

Understanding the Problem We are given a recursive function f ( n + 1 ) = f ( n ) − 2 and f ( t ) = 10 for some t . We want to find f ( 3 ) . The recursive function tells us that each term in the sequence is 2 less than the previous term.

Expressing f(3) in terms of f(t) We can express f ( 3 ) in terms of f ( t ) . If 3"> t > 3 , then to get from f ( t ) to f ( 3 ) , we need to add 2 for each step we go back. So f ( 3 ) = f ( t ) + 2 ( t − 3 ) . If t < 3 , then to get from f ( t ) to f ( 3 ) , we need to subtract 2 for each step we go forward. So f ( 3 ) = f ( t ) − 2 ( 3 − t ) = f ( t ) + 2 ( t − 3 ) . Therefore, in either case, f ( 3 ) = f ( t ) − 2 ( 3 − t ) .

Substituting f(t) = 10 Since f ( t ) = 10 , we have f ( 3 ) = 10 − 2 ( 3 − t ) . We don't know the value of t , so we can't determine a unique numerical value for f ( 3 ) . However, we can test the answer choices to see if any of them can be obtained for some value of t .

Testing the answer choices If f ( 3 ) = 1 , then 1 = 10 − 2 ( 3 − t ) , so − 9 = − 2 ( 3 − t ) , which means 4.5 = 3 − t , so t = − 1.5 .
If f ( 3 ) = 6 , then 6 = 10 − 2 ( 3 − t ) , so − 4 = − 2 ( 3 − t ) , which means 2 = 3 − t , so t = 1 .
If f ( 3 ) = 8 , then 8 = 10 − 2 ( 3 − t ) , so − 2 = − 2 ( 3 − t ) , which means 1 = 3 − t , so t = 2 .
If f ( 3 ) = 30 , then 30 = 10 − 2 ( 3 − t ) , so 20 = − 2 ( 3 − t ) , which means − 10 = 3 − t , so t = 13 .

Finding a Possible Value for f(3) Since t can take any value, all the answer choices are possible. However, the problem is likely intended to have a specific value for f ( 3 ) . Let's assume that t = 3 . Then f ( 3 ) = 10 . This is not one of the answer choices. Let's assume that t = 4 . Then f ( 4 ) = 10 , and f ( 3 ) = f ( 4 ) + 2 = 10 + 2 = 12 . This is not one of the answer choices. Let's assume that t = 2 . Then f ( 2 ) = 10 , and f ( 3 ) = f ( 2 ) − 2 = 10 − 2 = 8 . Therefore, f ( 3 ) = 8 is a possible value.

Assuming t=2 If we assume that n and t are close, let's consider the case where t = 2 . Then f ( 2 ) = 10 , and since f ( 3 ) = f ( 2 ) − 2 , we have f ( 3 ) = 10 − 2 = 8 .

Final Answer Therefore, a possible value for f ( 3 ) is 8.


Examples
Recursive functions are used in computer science to define functions in terms of themselves. For example, the factorial function can be defined recursively as n ! = n × ( n − 1 )! , with the base case 0 ! = 1 . This means that to calculate the factorial of a number, you multiply the number by the factorial of the number minus one, until you reach the base case. This concept is also used in finance to calculate compound interest, where the interest earned in each period is added to the principal, and the new amount earns interest in the next period.

Answered by GinnyAnswer | 2025-07-03

By analyzing the recursive function and substituting the known value f ( t ) = 10 , we find that a possible value for f ( 3 ) is 8 when assuming t = 2 .
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Answered by Anonymous | 2025-07-04