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A babysitter charges $30 for the first hour of babysitting, and the hourly rate is halved for each additional hour. Write a geometric series model using sigma notation to represent the income of the babysitter after 8 hours. Use the model you created to find how much the babysitter’s total pay after 8 hours.

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To represent the income of the babysitter after 8 hours, we can break down the charges as follows:

  1. For the first hour, the charge is $30.
  2. For each subsequent hour, the hourly rate is halved. Thus, for the second hour, it will be $15, for the third hour it will be $7.50, and so on.

We can see that after the first hour, the earnings form a geometric series with a first term of $15 and a common ratio of \( \frac{1}{2} \):

The hourly rates for the subsequent hours will be:

  • 2nd hour: \( 30 \times \left(\frac{1}{2}\right) = 15 \)
  • 3rd hour: \( 30 \times \left(\frac{1}{2}\right)^2 = 7.5 \)
  • 4th hour: \( 30 \times \left(\frac{1}{2}\right)^3 = 3.75 \)
  • 5th hour: \( 30 \times \left(\frac{1}{2}\right)^4 = 1.875 \)
  • 6th hour: \( 30 \times \left(\frac{1}{2}\right)^5 = 0.9375 \)
  • 7th hour: \( 30 \times \left(\frac{1}{2}\right)^6 = 0.46875 \)
  • 8th hour: \( 30 \times \left(\frac{1}{2}\right)^7 = 0.234375 \)

So, we can express the total income after 8 hours as a series:

\[ \text{Total Income} = 30 + \sum_{n=1}^{7} 30 \left(\frac{1}{2}\right)^n \]

This is a geometric series where:

  • The first term \( a = 30 \times \frac{1}{2} = 15 \)
  • The common ratio \( r = \frac{1}{2} \)
  • The number of terms, excluding the first hour, is 7.

The sum of a geometric series can be calculated with the formula:

\[ S_n = a \frac{1 - r^n}{1 - r} \]

For our series:

  • The first term of the series (after the first hour) is \( a = 15 \).
  • The common ratio is \( r = \frac{1}{2} \).
  • The number of terms is \( n = 7 \).

Now we can calculate the sum of this series:

\[ S_7 = 15 \frac{1 - \left(\frac{1}{2}\right)^7}{1 - \frac{1}{2}} \] \[ = 15 \frac{1 - \frac{1}{128}}{\frac{1}{2}} \] \[ = 15 \times 2 \left(1 - \frac{1}{128}\right) \] \[ = 30 \left(\frac{128 - 1}{128}\right) \] \[ = 30 \left(\frac{127}{128}\right) \] \[ = \frac{3810}{128} \] \[ \approx 29.765625 \]

Finally, we add the earnings from the first hour:

\[ \text{Total Income} = 30 + 29.765625 = 59.765625 \]

Rounding to two decimal places, the total income after 8 hours is approximately $59.77.