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To solve this problem, we need to find the total amount of money after 30 days by evaluating and multiplying the sums of three given geometric series:
[tex]\[ \begin{array}{r} \sum_{n=1}^{30} (1) \cdot (2)^{n-1} \\ \times \quad \sum_{n=1}^{29} (1) \cdot (2)^{n-1} \\ \times \quad \sum_{n=1}^{30} (1) \cdot \left(\frac{1}{2}\right)^{n-1} \end{array} \][/tex]
First, we’ll evaluate each series separately.
### First Geometric Series
The first series is:
[tex]\[ \sum_{n=1}^{30} 2^{n-1} \][/tex]
This is a geometric series where the first term [tex]\( a = 1 \)[/tex] (since [tex]\( 2^{0} = 1 \)[/tex]) and the common ratio [tex]\( r = 2 \)[/tex]. The sum [tex]\( S \)[/tex] of the first [tex]\( k \)[/tex] terms of a geometric series can be found using the formula:
[tex]\[ S = a \frac{r^k - 1}{r - 1} \][/tex]
For our series:
- [tex]\( a = 1 \)[/tex]
- [tex]\( r = 2 \)[/tex]
- [tex]\( k = 30 \)[/tex]
Substituting these values, we get:
[tex]\[ S_1 = 1 \frac{2^{30} - 1}{2 - 1} = 2^{30} - 1 = 1073741823 \][/tex]
### Second Geometric Series
The second series is:
[tex]\[ \sum_{n=1}^{29} 2^{n-1} \][/tex]
Similar to the first series, where:
- [tex]\( a = 1 \)[/tex]
- [tex]\( r = 2 \)[/tex]
- [tex]\( k = 29 \)[/tex]
Using the geometric series sum formula again, we get:
[tex]\[ S_2 = 1 \frac{2^{29} - 1}{2 - 1} = 2^{29} - 1 = 536870911 \][/tex]
### Third Geometric Series
The third series is:
[tex]\[ \sum_{n=1}^{30} \left(\frac{1}{2}\right)^{n-1} \][/tex]
For this series:
- [tex]\( a = 1 \)[/tex] (since [tex]\( \left(\frac{1}{2}\right)^0 = 1 \)[/tex])
- [tex]\( r = \frac{1}{2} \)[/tex]
- [tex]\( k = 30 \)[/tex]
Using the geometric series sum formula, we get:
[tex]\[ S_3 = 1 \frac{\left(\frac{1}{2}\right)^{30} - 1}{\left(\frac{1}{2}\right) - 1} = \frac{1 - \left(\frac{1}{2}\right)^{30}}{1 - \frac{1}{2}} = 2 \left(1 - \left(\frac{1}{2}\right)^{30}\right) \][/tex]
Since [tex]\( \left(\frac{1}{2}\right)^{30} \)[/tex] is a very small number, approximating it to near zero for practical purposes, we get:
[tex]\[ S_3 \approx 2 \][/tex]
Combining these results:
[tex]\[ S_1 = 1073741823 \][/tex]
[tex]\[ S_2 = 536870911 \][/tex]
[tex]\[ S_3 \approx 2 \][/tex]
### Product of the Sums
The total amount of money after 30 days is the product of these sums:
[tex]\[ \text{Total Money} = S_1 \times S_2 \times S_3 = 1073741823 \times 536870911 \times 2 \][/tex]
### Calculating the Total Money
From the provided result, we know:
[tex]\[ \text{Total Money} = 1.1529215003118797 \times 10^{18} \][/tex]
Thus, after 30 days, the total amount of money is:
[tex]\[ \boxed{1.1529215003118797 \times 10^{18}} \][/tex]
[tex]\[ \begin{array}{r} \sum_{n=1}^{30} (1) \cdot (2)^{n-1} \\ \times \quad \sum_{n=1}^{29} (1) \cdot (2)^{n-1} \\ \times \quad \sum_{n=1}^{30} (1) \cdot \left(\frac{1}{2}\right)^{n-1} \end{array} \][/tex]
First, we’ll evaluate each series separately.
### First Geometric Series
The first series is:
[tex]\[ \sum_{n=1}^{30} 2^{n-1} \][/tex]
This is a geometric series where the first term [tex]\( a = 1 \)[/tex] (since [tex]\( 2^{0} = 1 \)[/tex]) and the common ratio [tex]\( r = 2 \)[/tex]. The sum [tex]\( S \)[/tex] of the first [tex]\( k \)[/tex] terms of a geometric series can be found using the formula:
[tex]\[ S = a \frac{r^k - 1}{r - 1} \][/tex]
For our series:
- [tex]\( a = 1 \)[/tex]
- [tex]\( r = 2 \)[/tex]
- [tex]\( k = 30 \)[/tex]
Substituting these values, we get:
[tex]\[ S_1 = 1 \frac{2^{30} - 1}{2 - 1} = 2^{30} - 1 = 1073741823 \][/tex]
### Second Geometric Series
The second series is:
[tex]\[ \sum_{n=1}^{29} 2^{n-1} \][/tex]
Similar to the first series, where:
- [tex]\( a = 1 \)[/tex]
- [tex]\( r = 2 \)[/tex]
- [tex]\( k = 29 \)[/tex]
Using the geometric series sum formula again, we get:
[tex]\[ S_2 = 1 \frac{2^{29} - 1}{2 - 1} = 2^{29} - 1 = 536870911 \][/tex]
### Third Geometric Series
The third series is:
[tex]\[ \sum_{n=1}^{30} \left(\frac{1}{2}\right)^{n-1} \][/tex]
For this series:
- [tex]\( a = 1 \)[/tex] (since [tex]\( \left(\frac{1}{2}\right)^0 = 1 \)[/tex])
- [tex]\( r = \frac{1}{2} \)[/tex]
- [tex]\( k = 30 \)[/tex]
Using the geometric series sum formula, we get:
[tex]\[ S_3 = 1 \frac{\left(\frac{1}{2}\right)^{30} - 1}{\left(\frac{1}{2}\right) - 1} = \frac{1 - \left(\frac{1}{2}\right)^{30}}{1 - \frac{1}{2}} = 2 \left(1 - \left(\frac{1}{2}\right)^{30}\right) \][/tex]
Since [tex]\( \left(\frac{1}{2}\right)^{30} \)[/tex] is a very small number, approximating it to near zero for practical purposes, we get:
[tex]\[ S_3 \approx 2 \][/tex]
Combining these results:
[tex]\[ S_1 = 1073741823 \][/tex]
[tex]\[ S_2 = 536870911 \][/tex]
[tex]\[ S_3 \approx 2 \][/tex]
### Product of the Sums
The total amount of money after 30 days is the product of these sums:
[tex]\[ \text{Total Money} = S_1 \times S_2 \times S_3 = 1073741823 \times 536870911 \times 2 \][/tex]
### Calculating the Total Money
From the provided result, we know:
[tex]\[ \text{Total Money} = 1.1529215003118797 \times 10^{18} \][/tex]
Thus, after 30 days, the total amount of money is:
[tex]\[ \boxed{1.1529215003118797 \times 10^{18}} \][/tex]
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