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Emma earns a
$39,000 salary in the first year of her career. Each year, she gets a
5% raise.
How much does Emma earn in total in the first 11 years of her career? Round your final answer to the nearest thousand.

Emma earns a $\$ 39,000$ salary in the first year of her career. Each year, she gets a $5 \%$ raise. $\newline$ How much does Emma earn in total in the first $11$ years of her career? Round your final answer to the nearest thousand.

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Exponential growth and decay: word problems

Full solution

Q. Emma earns a

\$ 39,000

salary in the first year of her career. Each year, she gets a

5 \%

raise.

\newline

How much does Emma earn in total in the first

11

years of her career? Round your final answer to the nearest thousand.

Identify initial salary: Identify the initial salary and the annual raise percentage. $\newline$ Emma's initial salary is $\$39,000$ , and she gets a $5\%$ raise each year.
Calculate salary for each year: Calculate the salary for each year using the formula for the nth term of a geometric sequence. $\newline$ The formula for the nth term of 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. $\newline$ In this case, $a_1 = \$(39,000)$ and $r = 1 + \frac{5}{100} = 1.05$ .
Calculate total salary: Calculate the total salary over $11$ years. $\newline$ The total salary over $11$ years is the sum of a geometric series, which can be calculated using the formula $S_n = a_1 \times (1 - r^n) / (1 - r)$ , where $S_n$ is the sum of the first $n$ terms. $\newline$ Here, $n = 11$ , so we need to calculate $S_{11}$ .
Substitute values into formula: Substitute the values into the formula to calculate the total salary. $\newline$ $S_{11} = \$39,000 \times (1 - 1.05^{11}) / (1 - 1.05)$
Calculate total salary: Calculate the total salary using the values from Step $4$ . $\newline$ $S_{11} = (\$)39,000 \times (1 - 1.05^{11}) / (1 - 1.05)$ $\newline$ $S_{11} = (\$)39,000 \times (1 - 1.71034) / (-0.05)$ $\newline$ $S_{11} = (\$)39,000 \times (-0.71034) / (-0.05)$ $\newline$ $S_{11} = (\$)39,000 \times 14.2068$ $\newline$ $S_{11} = (\$)554,065.2$
Round final answer: Round the final answer to the nearest thousand. The total salary rounded to the nearest thousand is approximately $\$554,000$ .

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