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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. dollars

Emma earns a $39,000 \$ 39,000 salary in the first year of her career. Each year, she gets a 5% 5 \% raise.\newlineHow much does Emma earn in total in the first 1111 years of her career? Round your final answer to the nearest thousand.\newlinedollars

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Q. Emma earns a $39,000 \$ 39,000 salary in the first year of her career. Each year, she gets a 5% 5 \% raise.\newlineHow much does Emma earn in total in the first 1111 years of her career? Round your final answer to the nearest thousand.\newlinedollars
  1. Initial salary and raise percentage: Emma earns a $39,000\$39,000 salary in the first year of her career. Each year, she gets a 5%5\% raise. To find out how much Emma earns in total in the first 1111 years of her career, we need to calculate the sum of a geometric series where the first term is her initial salary and the common ratio is the annual raise percentage converted to a multiplier.
  2. Formula for calculating the sum of a geometric series: The first term aa of the geometric series is her initial salary, which is $39,000\$39,000. The common ratio rr is 11 plus the raise percentage expressed as a decimal. Since the raise is 5%5\%, the common ratio is 1+0.05=1.051 + 0.05 = 1.05.
  3. Substituting values into the formula: The sum of the first nn terms of a geometric series can be calculated using the formula Sn=a(1−rn)/(1−r)S_n = a(1 - r^n) / (1 - r), where SnS_n is the sum of the first nn terms, aa is the first term, rr is the common ratio, and nn is the number of terms. In this case, nn is 1111 because we are looking at the first 1111 years of her career.
  4. Calculating the value of 1.05111.05^{11}: Substitute the known values into the formula to calculate the total earnings over the 1111 years: S11=39000(1−1.0511)/(1−1.05)S_{11} = 39000(1 - 1.05^{11}) / (1 - 1.05).
  5. Calculating the numerator: Calculate the sum: S11=39000(1−1.0511)(1−1.05)S_{11} = \frac{39000(1 - 1.05^{11})}{(1 - 1.05)}. First, calculate the value of 1.05111.05^{11}.
  6. Calculating the denominator: 1.05111.05^{11} is approximately 1.710341.71034. Now substitute this value back into the sum formula: S11=39000(1−1.71034)/(1−1.05)S_{11} = 39000(1 - 1.71034) / (1 - 1.05).
  7. Finding the sum: Calculate the numerator: 39000(1−1.71034)=39000(−0.71034)=−27683.2639000(1 - 1.71034) = 39000(-0.71034) = -27683.26. This is a negative number because we subtracted a number greater than 11 from 11, which represents the total growth factor over 1111 years.
  8. Rounding the final answer: Now calculate the denominator: 1−1.05=−0.051 - 1.05 = -0.05. This is also a negative number, representing the growth factor per year subtracted from 11.
  9. Rounding the final answer: Now calculate the denominator: 1−1.05=−0.051 - 1.05 = -0.05. This is also a negative number, representing the growth factor per year subtracted from 11.Divide the numerator by the denominator to find the sum: S11=−27683.26/−0.05=553665.2S_{11} = -27683.26 / -0.05 = 553665.2.
  10. Rounding the final answer: Now calculate the denominator: 1−1.05=−0.051 - 1.05 = -0.05. This is also a negative number, representing the growth factor per year subtracted from 11.Divide the numerator by the denominator to find the sum: S11=−27683.26/−0.05=553665.2S_{11} = -27683.26 / -0.05 = 553665.2.Round the final answer to the nearest thousand: $S_{\(11\)} \approx \$(\(554\),\(000\)).

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