If [tex]f(n+1)=f(n)-2[/tex] and [tex]f(1)=10[/tex], what is [tex]f(3)[/tex]?
A. 1
B. 6
C. 8
D. 30
Answer (2)
We are given the recursive formula f ( n + 1 ) = f ( n ) − 2 and f ( 1 ) = 10 .
We find f ( 2 ) by substituting n = 1 into the formula: f ( 2 ) = f ( 1 ) − 2 = 10 − 2 = 8 .
We find f ( 3 ) by substituting n = 2 into the formula: f ( 3 ) = f ( 2 ) − 2 = 8 − 2 = 6 .
Therefore, f ( 3 ) = 6 .
Explanation
Understanding the Problem We are given a recursive formula f ( n + 1 ) = f ( n ) − 2 and the initial condition f ( 1 ) = 10 . Our goal is to find the value of f ( 3 ) . This means we need to apply the recursive formula twice, first to find f ( 2 ) and then to find f ( 3 ) .
Finding f(2) First, let's find f ( 2 ) . We substitute n = 1 into the recursive formula: f ( 1 + 1 ) = f ( 1 ) − 2
f ( 2 ) = f ( 1 ) − 2 Since f ( 1 ) = 10 , we have f ( 2 ) = 10 − 2 = 8
Finding f(3) Now, let's find f ( 3 ) . We substitute n = 2 into the recursive formula: f ( 2 + 1 ) = f ( 2 ) − 2 f ( 3 ) = f ( 2 ) − 2 Since f ( 2 ) = 8 , we have f ( 3 ) = 8 − 2 = 6
Final Answer Therefore, the value of f ( 3 ) is 6.
Examples
Recursive formulas are used in many real-world scenarios, such as calculating compound interest, modeling population growth, and predicting the spread of diseases. For example, if you deposit money into a bank account that earns interest each year, the balance in your account can be modeled using a recursive formula. Similarly, the number of bacteria in a petri dish can be modeled using a recursive formula if the bacteria reproduce at a constant rate.
By using the recursive relation f ( n + 1 ) = f ( n ) − 2 starting from f ( 1 ) = 10 , we find that f ( 2 ) = 8 and subsequently f ( 3 ) = 6 . Hence, the answer is B. 6.
;
