Question
A computer programmer makes $60,000 in her first year of working at a company. She gets a 10 percent pay raise every year. Create a geometric series model for how much she makes in his first 5 years of working.(1 point)
Responses
∑5n=360,000(1.1)n−1
sigma-summation underscript n equals 3 overscript 5 End Scripts 60,000 left parenthesis 1.1 right parenthesis superscript n minus 1 baseline
∑5n=160,000(0.1)n−1
sigma-summation underscript n equals 1 overscript 5 End Scripts 60,000 left parenthesis 0.1 right parenthesis superscript n minus 1 baseline
∑5n=560,000(1.1)n
sigma-summation underscript n equals 5 overscript 5 End Scripts 60,000 left parenthesis 1.1 right parenthesis superscript n baseline
∑5n=160,000(1.1)n−1
sigma-summation underscript n equals 1 overscript 5 End Scripts 60,000 left parenthesis 1.1 right parenthesis superscript n minus 1 baseline
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Responses
∑5n=360,000(1.1)n−1
sigma-summation underscript n equals 3 overscript 5 End Scripts 60,000 left parenthesis 1.1 right parenthesis superscript n minus 1 baseline
∑5n=160,000(0.1)n−1
sigma-summation underscript n equals 1 overscript 5 End Scripts 60,000 left parenthesis 0.1 right parenthesis superscript n minus 1 baseline
∑5n=560,000(1.1)n
sigma-summation underscript n equals 5 overscript 5 End Scripts 60,000 left parenthesis 1.1 right parenthesis superscript n baseline
∑5n=160,000(1.1)n−1
sigma-summation underscript n equals 1 overscript 5 End Scripts 60,000 left parenthesis 1.1 right parenthesis superscript n minus 1 baseline
Skip to navigation
Answers
GPT-4o mini
To create a geometric series model for the programmer's salary over her first 5 years, we need to understand how her salary increases each year due to the 10% increase.
Starting with a salary of \(60,000\), her salary for each year can be calculated as follows:
- Year 1: \(60,000\)
- Year 2: \(60,000 \times 1.1 = 66,000\)
- Year 3: \(60,000 \times (1.1)^2 = 72,600\)
- Year 4: \(60,000 \times (1.1)^3 = 79,860\)
- Year 5: \(60,000 \times (1.1)^4 = 87,846\)
To express her total earnings over the first 5 years as a geometric series, we will use the formula for the sum of a geometric series:
\[
S_n = a \frac{(r^n - 1)}{(r - 1)}
\]
where:
- \(S_n\) is the sum of the first \(n\) terms,
- \(a\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the number of terms.
In this case:
- \(a = 60,000\),
- \(r = 1.1\),
- \(n = 5\).
Putting these values into the formula gives us:
\[
S_5 = 60,000 \frac{(1.1^5 - 1)}{(1.1 - 1)}
\]
However, you can also represent the sum as a summation notation:
\[
\sum_{n=1}^{5} 60,000 (1.1)^{n-1}
\]
Given the options provided, the correct representation of her salary over the first 5 years as a geometric series is:
\[
\sum_{n=1}^{5} 60,000 (1.1)^{n-1}
\]
So the correct option is:
**\(\sum_{n=1}^{5} 60,000 (1.1)^{n-1}\)**
Starting with a salary of \(60,000\), her salary for each year can be calculated as follows:
- Year 1: \(60,000\)
- Year 2: \(60,000 \times 1.1 = 66,000\)
- Year 3: \(60,000 \times (1.1)^2 = 72,600\)
- Year 4: \(60,000 \times (1.1)^3 = 79,860\)
- Year 5: \(60,000 \times (1.1)^4 = 87,846\)
To express her total earnings over the first 5 years as a geometric series, we will use the formula for the sum of a geometric series:
\[
S_n = a \frac{(r^n - 1)}{(r - 1)}
\]
where:
- \(S_n\) is the sum of the first \(n\) terms,
- \(a\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the number of terms.
In this case:
- \(a = 60,000\),
- \(r = 1.1\),
- \(n = 5\).
Putting these values into the formula gives us:
\[
S_5 = 60,000 \frac{(1.1^5 - 1)}{(1.1 - 1)}
\]
However, you can also represent the sum as a summation notation:
\[
\sum_{n=1}^{5} 60,000 (1.1)^{n-1}
\]
Given the options provided, the correct representation of her salary over the first 5 years as a geometric series is:
\[
\sum_{n=1}^{5} 60,000 (1.1)^{n-1}
\]
So the correct option is:
**\(\sum_{n=1}^{5} 60,000 (1.1)^{n-1}\)**