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 in the sequence is 3 .
Explanation
Understanding the Problem We are given a sequence defined recursively by the formula f ( n + 1 ) = − 2 f ( n ) . The first term of the sequence is f ( 1 ) = − 1.5 . We want to find the next term in the sequence, which is f ( 2 ) .
Applying the Recursive Formula To find the next term, we can use the recursive formula. Let n = 1 . Then, we have f ( 1 + 1 ) = f ( 2 ) = − 2 f ( 1 ) .
Substituting the Value of f(1) We know that f ( 1 ) = − 1.5 . Substituting this value into the equation, we get f ( 2 ) = − 2 × ( − 1.5 ) .
Calculating f(2) Now, we calculate the value of f ( 2 ) : f ( 2 ) = − 2 × ( − 1.5 ) = 3 .
Final Answer Therefore, the next term in the sequence is 3.
Examples
Recursive sequences are used in various real-life applications, such as modeling population growth, calculating compound interest, and designing fractals. For example, if a population doubles every year, we can model it using a recursive sequence where each term is twice the previous term. Similarly, compound interest can be calculated recursively, where the interest earned in each period is added to the principal to calculate the new balance. Understanding recursive sequences helps in predicting future values and making informed decisions in these scenarios.
The next term in the sequence is 3, calculated using the formula f ( n + 1 ) = − 2 f ( n ) with the first term f ( 1 ) = − 1.5 . By substituting n = 1 , we found f ( 2 ) = − 2 × ( − 1.5 ) = 3 .
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