Answered

An electrician earns $110 after his first hour of working for a client. His total pay based on the number of hours worked can be represented using the sequence shown.

$110, 130, 150, 170, \ldots$



Which recursive formula can be used to determine the total amount of money earned for each successive hour worked based on the amount of money currently earned?

A$f(n+1)=f(n)+20$
B$f(n+1)=f(n)+110$
C$f(n+1)=f(n+1)+20$
D$f(n+1)=f(n+1)+110$

Verified Answer :

  • Identify the sequence as an arithmetic progression with a constant increase of \(\$20\) between each hour.
  • Use the general recursive form for an arithmetic sequence: \(f(n+1) = f(n) + d\).
  • Substitute the common difference \(d = 20\) into the formula to obtain \(f(n+1) = f(n) + 20\).
  • The correct recursive formula is \(\boxed{f(n+1) = f(n) + 20}\).

Explanation

  1. Analysis of the Sequence

The sequence represents the total pay of an electrician after each successive hour of work: \(110, 130, 150, 170, \dots\). To determine the recursive formula, we first identify the type of sequence. By observing the difference between consecutive terms:
\(130 - 110 = 20\)
\(150 - 130 = 20\)
\(170 - 150 = 20\)
Since the difference between any two consecutive terms is constant, this is an arithmetic sequence with a common difference \(d = 20\).

  1. Determining the Recursive Formula

A recursive formula for an arithmetic sequence defines the next term, \(f(n+1)\), in terms of the current term, \(f(n)\), and the common difference, \(d\). The general form is:
\[f(n+1) = f(n) + d\]
We have already determined that the common difference \(d\) is \(20\). By substituting this value into the general formula, we get:
\[f(n+1) = f(n) + 20\]
This formula indicates that the total pay after \(n+1\) hours is equal to the total pay after \(n\) hours plus the additional hourly rate of \(\$20\).

  1. Selecting the Correct Formula

Comparing our derived formula \(f(n+1) = f(n) + 20\) with the provided options:

  • \(f(n+1) = f(n) + 20\) (Matches)
  • \(f(n+1) = f(n) + 110\)
  • \(f(n+1) = f(n+1) + 20\)
  • \(f(n+1) = f(n+1) + 110\)

The first option is the correct representation of the sequence. Therefore, the recursive formula is \(\boxed{f(n+1) = f(n) + 20}\).

Examples

Understanding recursive formulas is essential for modeling real-world processes that occur in steps, such as calculating compound interest on a savings account, predicting population growth over time, or managing costs that increase incrementally. In this scenario, the electrician's income is modeled by a linear increase per hour, which is a foundational concept in financial planning and project estimation. By recognizing the pattern as an arithmetic progression, one can easily forecast earnings for any number of future hours worked.

Related Questions