Question

extra practice:
1.

	linear	exponential	neither

2.
tennis tournament

rounds	1	2	3	4	5

there are 4 players remaining after 5 rounds.

	linear	exponential	neither

1. after graduating from vanderbilt university, janee has two job offers to consider.

• a consulting firm offers her $90,000 per year but with no hope of ever having an increase in pay. janee knows that very few of her peers are getting that kind of an offer right out of college, and it sounds very tempting.
• a social media start - up also offers her a job, but guarantees a mere $25,000 per year. janee would have to get a roommate and wait to pay off her student loans in order to make this job work, but the founder is sure that he’ll be able to go public and offers her a 15% raise at the beginning of each new year.

a) which job will have a greater annual salary at the beginning of the fifth year? by approximately how much?
b) which job will have a greater annual salary at the beginning of the tenth year? by approximately how much?
c) which job will have a greater annual salary at the beginning of the twentieth year? by approximately how much?
d) if you were in jane’s shoes, which job would you take?

Explanation:

Step1: Determine the salary formula for each job

• Consulting firm: The salary \(C(n)=90000\) for all \(n\) (where \(n\) is the year).
• Social - media start - up: The salary is a geometric sequence. The formula for a geometric sequence is \(S(n)=a\times r^{n - 1}\), where \(a = 25000\) (initial salary) and \(r=1 + 0.15=1.15\). So \(S(n)=25000\times(1.15)^{n - 1}\)

Step2: Calculate the salaries for each part

Part (a)

• Consulting firm: \(C(5)=90000\)
• Social - media start - up: \(S(5)=25000\times(1.15)^{5 - 1}=25000\times(1.15)^{4}\)

$$
LATEXBLOCK0
$$

\(S(5)=25000\times1.74900625 = 43725.16\)

• Difference: \(90000-43725.16 = 46274.84\)

Part (b)

• Consulting firm: \(C(10)=90000\)
• Social - media start - up: \(S(10)=25000\times(1.15)^{10 - 1}=25000\times(1.15)^{9}\)

$$
LATEXBLOCK1
$$

\(S(10)=25000\times3.517877=87946.93\)

• Difference: \(90000 - 87946.93=2053.07\)

Part (c)

• Consulting firm: \(C(20)=90000\)
• Social - media start - up: \(S(20)=25000\times(1.15)^{20 - 1}=25000\times(1.15)^{19}\)

$$
LATEXBLOCK2
$$

\(S(20)=25000\times16.36653 = 409163.25\)

• Difference: \(409163.25-90000=319163.25\)

Answer:

a) The consulting firm. The consulting - firm salary is approximately \(\$46274.84\) greater.
b) The consulting firm. The consulting - firm salary is approximately \(\$2053.07\) greater.
c) The social - media start - up. The social - media start - up salary is approximately \(\$319163.25\) greater.
d) This is a subjective answer. If one values short - term financial stability, the consulting firm is a good choice. If one is willing to take a risk for potentially higher long - term earnings, the social - media start - up may be better.