A sequence is defined by the recursive function $f(n+1)=f(n)-2$. If $f(1)=10$, what is $f(3)$?
Answer (2)
6
Explanation
Understanding the Problem We are given a recursive sequence defined by f ( n + 1 ) = f ( n ) − 2 and the initial value f ( 1 ) = 10 . Our goal is to find the value of f ( 3 ) .
Finding f(2) To find f ( 3 ) , we first need to find f ( 2 ) using the recursive formula. Substituting n = 1 into the formula, we get f ( 1 + 1 ) = f ( 1 ) − 2 , which simplifies to f ( 2 ) = f ( 1 ) − 2 . Since f ( 1 ) = 10 , we have f ( 2 ) = 10 − 2 = 8 .
Finding f(3) Now that we have f ( 2 ) = 8 , we can find f ( 3 ) using the recursive formula again. Substituting n = 2 into the formula, we get f ( 2 + 1 ) = f ( 2 ) − 2 , which simplifies to 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 sequences are used in many real-world applications, such as calculating compound interest, modeling population growth, and designing computer algorithms. For example, if you deposit money into a bank account that earns interest each year, the amount of money in your account can be modeled by a recursive sequence. Understanding recursive sequences helps in predicting future values based on current values and a defined pattern.
To find f ( 3 ) , we first calculate f ( 2 ) using the formula f ( 2 ) = 10 − 2 = 8 , and then use f ( 3 ) = f ( 2 ) − 2 to find that f ( 3 ) = 6 .
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