Clark puts money into his savings account for a week. The first day he puts $200 in, and then each subsequent day he puts in half of what he puts in half of what he put in the day before. Create a geometric series to model how money Clark puts into savings. Use this geometric series to calculate how much money he puts into his savings account:
The geometric series can be modeled using the formula:
\[a_n = a_1 \cdot r^{n-1}\]
where:
- \(a_1 = $200\) (first day)
- \(r = 0.5\) (half of the previous day's amount)
- \(n\) is the number of days
Using the formula, the amount of money Clark puts into his savings account on the nth day can be calculated as:
\[a_n = 200 \cdot (0.5)^{n-1}\]
To calculate how much money he puts into his savings account for a week (7 days), you need to calculate the sum of this geometric series:
\[S = a_1 \cdot \frac{1-r^n}{1-r}\]
\[S = 200 \cdot \frac{1-(0.5)^7}{1-0.5}\]
\[S = 200 \cdot \frac{1-0.0078125}{0.5}\]
\[S = 200 \cdot \frac{0.9921875}{0.5}\]
\[S = 200 \cdot 1.984375\]
\[S = $396.875\]
Therefore, Clark puts a total of $396.875 into his savings account for a week.