Question

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)
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:

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