To determine the value of the fifth term in a series where each term is 10% greater than the previous term, and the first term is 50, follow these steps: 1. **Identify the Series Type**: This series is a geometric sequence, where each term is a fixed percentage greater than the previous term. The coRead more
To determine the value of the fifth term in a series where each term is 10% greater than the previous term, and the first term is 50, follow these steps:
1. **Identify the Series Type**: This series is a geometric sequence, where each term is a fixed percentage greater than the previous term. The common ratio (r) is 1.10, which represents the 10% increase (100% + 10% = 110% = 1.10).
2. **Write the Formula for the \(n\)-th Term**: The formula for the \(n\)-th term \(a_n\) in a geometric sequence is:
\[
a_n = a_1 \times r^{(n-1)}
\]
where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
3. **Substitute the Values**: For the fifth term (\(n = 5\)), \(a_1 = 50\) and \(r = 1.10\). Plug these values into the formula:
\[
a_5 = 50 \times (1.10)^{(5-1)}
\]
\[
a_5 = 50 \times (1.10)^4
\]
4. **Calculate \( (1.10)^4 \)**:
\[
(1.10)^4 \approx 1.4641
\]
5. **Find the Fifth Term**:
\[
a_5 = 50 \times 1.4641 \approx 73.21
\]
Thus, the value of the fifth term is approximately $73.21.
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To determine the total amount of the investment given that the average annual interest earned is $200, and this represents 5% of the total investment, follow these steps: 1. **Define the Variables**: Let \( P \) be the total amount of the investment. 2. **Set Up the Equation**: We know that 5% of \(Read more
To determine the total amount of the investment given that the average annual interest earned is $200, and this represents 5% of the total investment, follow these steps:
1. **Define the Variables**: Let \( P \) be the total amount of the investment.
2. **Set Up the Equation**: We know that 5% of \( P \) equals $200. In mathematical terms, this can be written as:
\[
0.05 \times P = 200
\]
3. **Solve for \( P \)**: To find \( P \), divide both sides of the equation by 0.05:
\[
P = \frac{200}{0.05}
\]
4. **Calculate the Total Investment**:
\[
P = \frac{200}{0.05} = 4000
\]
Therefore, the total amount of the investment is $4,000. This result confirms that if the interest earned ($200) is 5% of the total amount, then the total investment must be $4,000.
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