Each year, Francesca earns a salary that is 2% higher than her previous year's salary. In her first 5 years at this job, she earned a total of $187,345. What was Francesca's salary in her 1^("st ") year at this job? Round your final answer to the nearest thousand. dollars

Determine growth type: Determine the type of growth Francesca's salary undergoes. Francesca's salary increases by 2% each year, which is a form of exponential growth.Denote Francesca's salary: Let's denote Francesca's first year salary as S. Each subsequent year, her salary is 2% more than the previous year. This can be represented as S, S(1+0.02), S(1+0.02)^2, S(1+0.02)^3, and S(1+0.02)^4 for the first five years.Calculate total salary over 5 years: The total salary over the first five years is the sum of the geometric sequence: S + S(1+0.02) + S(1+0.02)^2 + S(1+0.02)^3 + S(1+0.02)^4 = $187,345.Solve equation for sum of geometric series: To find S, we need to solve the equation for the sum of a geometric series: S * [1 + (1+0.02) + (1+0.02)^2 + (1+0.02)^3 + (1+0.02)^4] = $187,345.Calculate sum of geometric factors: Calculate the sum of the geometric factors: 1 + (1+0.02) + (1+0.02)^2 + (1+0.02)^3 + (1+0.02)^4. 1 + 1.02 + 1.02^2 + 1.02^3 + 1.02^4 = 1 + 1.02 + 1.0404 + 1.061208 + 1.08243264 ≈ 5.20404064.Find value of S: Now, divide the total salary by the sum of the geometric factors to find S: $187,345 / 5.20404064.Round final answer: Perform the division to find the value of S: $187,345 / 5.20404064 ≈ $35,994.82.Round the final answer to the nearest thousand as instructed: $35,994.82 rounded to the nearest thousand is $36,000.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Each year, Francesca earns a salary that is
2% higher than her previous year's salary. In her first 5 years at this job, she earned a total of
$187,345.
What was Francesca's salary in her
1^("st ") year at this job?
Round your final answer to the nearest thousand.
dollars

Each year, Francesca earns a salary that is $2 \%$ higher than her previous year's salary. In her first $5$ years at this job, she earned a total of $\$ 187,345$ . $\newline$ What was Francesca's salary in her $1^{\text {st }}$ year at this job? $\newline$ Round your final answer to the nearest thousand. $\newline$ dollars

View step-by-step help

Home

Math Problems

Algebra 1

Exponential growth and decay: word problems

Full solution

Q. Each year, Francesca earns a salary that is

2 \%

higher than her previous year's salary. In her first

5

years at this job, she earned a total of

\$ 187,345

\newline

What was Francesca's salary in her

1^{\text {st }}

year at this job?

\newline

Round your final answer to the nearest thousand.

\newline

dollars

Determine growth type: Determine the type of growth Francesca's salary undergoes. $\newline$ Francesca's salary increases by $2\%$ each year, which is a form of exponential growth.
Denote Francesca's salary: Let's denote Francesca's first year salary as $S$ . Each subsequent year, her salary is $2\%$ more than the previous year. This can be represented as $S$ , $S(1+0.02)$ , $S(1+0.02)^2$ , $S(1+0.02)^3$ , and $S(1+0.02)^4$ for the first five years.
Calculate total salary over $5$ years: The total salary over the first five years is the sum of the geometric sequence: $S + S(1+0.02) + S(1+0.02)^2 + S(1+0.02)^3 + S(1+0.02)^4 = \$(187,345).$
Solve equation for sum of geometric series: To find $S$ , we need to solve the equation for the sum of a geometric series: $S \times [$1$ + ($1$+$0$.$02$) + ($1$+$0$.$02$)^$2$ + ($1$+$0$.$02$)^$3$ + ($1$+$0$.$02$)^$4$] = \$$187$,$345$.
Calculate sum of geometric factors: Calculate the sum of the geometric factors: $1 + (1+0.02) + (1+0.02)^2 + (1+0.02)^3 + (1+0.02)^4$.$\newline$$1 + 1.02 + 1.02^2 + 1.02^3 + 1.02^4 = 1 + 1.02 + 1.0404 + 1.061208 + 1.08243264 \approx 5.20404064$.
Find value of S: Now, divide the total salary by the sum of the geometric factors to find S: $\$187,345 / 5.20404064$.
Round final answer: Perform the division to find the value of S: $\$187,345 / 5.20404064 \approx \$35,994.82$.
Round final answer: Perform the division to find the value of S: $\$187,345 / 5.20404064 \approx \$35,994.82$. Round the final answer to the nearest thousand as instructed: $\$35,994.82$ rounded to the nearest thousand is $\$36,000$.