A sequence is defined recursively by the formula [tex]$f(n+1)=-2 f(n)$[/tex]. The first term of the sequence is -1.5. What is the next term in the sequence?
Answer (2)
Use the recursive formula f ( n + 1 ) = − 2 f ( n ) .
Substitute n = 1 to find f ( 2 ) = − 2 f ( 1 ) .
Substitute f ( 1 ) = − 1.5 to get f ( 2 ) = − 2 ( − 1.5 ) .
Calculate f ( 2 ) = 3 , so the next term is 3 .
Explanation
Understanding the Problem We are given a recursive formula for a sequence: f ( n + 1 ) = − 2 f ( n ) . We also know that the first term of the sequence is f ( 1 ) = − 1.5 . Our goal is to find the next term in the sequence, which is f ( 2 ) .
Applying the Recursive Formula To find f ( 2 ) , we can use the recursive formula with n = 1 . This gives us f ( 1 + 1 ) = f ( 2 ) = − 2 f ( 1 ) .
Substituting the Value of f(1) Now, we substitute the value of f ( 1 ) into the equation: f ( 2 ) = − 2 ( − 1.5 ) .
Calculating f(2) Multiplying -2 by -1.5, we get f ( 2 ) = 3 .
Final Answer Therefore, the next term in the sequence is 3.
Examples
Recursive sequences are used in various fields, such as computer science (e.g., defining fractals), finance (e.g., modeling compound interest), and biology (e.g., population growth). For example, if a population doubles every year, we can model it with a recursive sequence. Understanding recursive sequences helps us predict future values based on previous ones.
The next term in the sequence defined by f ( n + 1 ) = − 2 f ( n ) with the first term f ( 1 ) = − 1.5 is 3 .
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