A photographer charges $100 for the first hour of a photoshoot, and the hourly rate is halved for each additional hour. Which of the following geometric series models represents the income of the photographer after 4 hours?(1 point)

Question

Responses

∑100n=14(12)n−1
sigma-summation underscript n equals 1 overscript 100 End Scripts 4 left parenthesis Start Fraction 1 over 2 End Fraction right parenthesis superscript n minus 1 baseline

∑4n=14(12)n−1
sigma-summation underscript n equals 1 overscript 4 End Scripts 4 left parenthesis Start Fraction 1 over 2 End Fraction right parenthesis superscript n minus 1 baseline

∑4n=112(100)n−1
sigma-summation underscript n equals 1 overscript 4 End Scripts Start Fraction 1 over 2 End Fraction left parenthesis 100 right parenthesis superscript n minus 1 baseline

∑4n=1100(12)n−1
sigma-summation underscript n equals 1 overscript 4 End Scripts 100 left parenthesis Start Fraction 1 over 2 End Fraction right parenthesis superscript n minus 1 baseline

Answer 1

To determine the correct geometric series that models the income of the photographer after 4 hours, we need to calculate the income for each of the 4 hours.

1. For the first hour, the charge is $100.
2. For the second hour, the charge is halved: \(\frac{100}{2} = $50\).
3. For the third hour, the charge is halved again: \(\frac{50}{2} = $25\).
4. For the fourth hour, the charge is halved once more: \(\frac{25}{2} = $12.50\).

Now, the income for each of these hours can be represented as:

• First hour: \(100 = 100(1)\)
• Second hour: \(50 = 100 \left(\frac{1}{2}\right)\)
• Third hour: \(25 = 100 \left(\frac{1}{2}\right)^2\)
• Fourth hour: \(12.50 = 100 \left(\frac{1}{2}\right)^3\)

The total income after 4 hours can be modeled as the sum of these terms:

\[ \text{Total Income} = 100 + 100 \left(\frac{1}{2}\right) + 100 \left(\frac{1}{2}\right)^2 + 100 \left(\frac{1}{2}\right)^3 \]

This can be expressed in a summation (sigma notation) as follows:

\[ \sum_{n=1}^{4} 100 \left(\frac{1}{2}\right)^{n-1} \]

Given the provided options:

1. \(\sum_{n=1}^{4} 100 \left(\frac{1}{2}\right)^{n-1}\)

This matches our derived sum perfectly.

Thus, the correct answer is:

\(\sum_{n=1}^{4} 100 \left(\frac{1}{2}\right)^{n-1}\).