Answered

Zina spends 1.5 hours setting up her sewing machine and making one hat. The total amount of time spent making hats can be represented by the sequence below.

$1.5, 2.25, 3.0, 3.75, \ldots$



Which recursive formula can be used to determine the total amount of time spent making hats based on the total amount of time spent previously?


A$f(n+1)=f(n)+1.5$
B$f(n+1)=f(n)+0.75$
C$f(n+1)=\frac{1}{2} f(n)$
D$f(n+1)=\frac{3}{2} f(n)$

Verified Answer :

  • Analyze the provided sequence \(1.5, 2.25, 3.0, 3.75, \ldots\) to identify the pattern between consecutive terms.
  • Calculate the constant difference between each term: \(2.25 - 1.5 = 0.75\), \(3.0 - 2.25 = 0.75\), and \(3.75 - 3.0 = 0.75\).
  • Recognize that a constant difference indicates an arithmetic sequence defined by the recursive rule \(f(n+1) = f(n) + d\).
  • Select the matching formula: \(\boxed{f(n+1) = f(n) + 0.75}\).

Explanation

  1. Analysis of the Sequence

The sequence given is \(1.5, 2.25, 3.0, 3.75, \ldots\). We are looking for a recursive formula \(f(n+1)\) in terms of \(f(n)\), where \(n\) is a positive integer representing the sequence position. Let's list the first few terms:

  • \(f(1) = 1.5\)
  • \(f(2) = 2.25\)
  • \(f(3) = 3.0\)
  • \(f(4) = 3.75\)
  1. Determining the Recursive Formula

To find a pattern, we examine the differences between consecutive terms:

  • \(f(2) - f(1) = 2.25 - 1.5 = 0.75\)
  • \(f(3) - f(2) = 3.0 - 2.25 = 0.75\)
  • \(f(4) - f(3) = 3.75 - 3.0 = 0.75\)

Since the difference between consecutive terms is a constant value of \(0.75\), the sequence is an arithmetic progression. In an arithmetic progression, the next term is obtained by adding the common difference \(d\) to the current term, which is given by the recursive formula:
\[f(n+1) = f(n) + d\]
Substituting our common difference \(d = 0.75\), we get:
\[f(n+1) = f(n) + 0.75\]

  1. Verification and Conclusion

Comparing our derived formula \(f(n+1) = f(n) + 0.75\) with the provided options:

  • \(f(n+1) = f(n) + 1.5\)
  • \(f(n+1) = f(n) + 0.75\)
  • \(f(n+1) = \frac{1}{2} f(n)\)
  • \(f(n+1) = \frac{3}{2} f(n)\)

The second option matches our result. Thus, the recursive formula that determines the total time spent is \(\boxed{f(n+1) = f(n) + 0.75}\).

Examples

In real life, recursive patterns like this are essential for project management and resource planning. If you are building multiple items, such as hats, furniture, or software modules, you often have a fixed setup time followed by a constant incremental production time for each additional unit. Understanding the recursive formula allows you to quickly project the total time required for \(n\) items without calculating each individual step, helping you manage deadlines and workload efficiently.

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