Question

1)Use the provided information to identify each of Mr. Nicholson’s earning opportunities as arithmetic or geometric. For each opportunity, include the common difference or ratio. In your final answer, use complete sentences to explain how you identified each opportunity as arithmetic or geometric.
2)Model each of Mr. Nicholson’s salary options with a recursive sequence that includes his potential earnings for the first three years of employment.
According to the first three terms of each sequence, can you conclude that there is a significant difference in Mr. Nicholson’s potential earnings with each increase option? Use complete sentences to explain your conclusion.

Answer 1

The missing provided information is that Mr. Nicholson accepts a job that pays an annual salary of $60,000. And he is given the option of choosing between two annual raises:

 a) an annual raise of $3,500 or b) an annual raise of 5% of his current salary.

Then, with that information you have to answer the given questions.Which I am going to do step by step.

1) identify each of Mr. Nicholson’s earning opportunities as arithmetic or geometric. For each opportunity, include the common difference or ratio. In your final answer, use complete sentences to explain how you identified each opportunity as arithmetic or geometric.

- An annual raise of fix $3500 means that every year the salary increase in a constant amount driving to this sequence:

60,000 + 3500 = 63,500;
63,500 + 3,500 =  67,000
67,000 + 3,500 = 70,500
70,500 + 3,500 = 74,000
74,000 + 3,500 = 77,500
...


Then you have a constant difference between two adjacent terms which means that this is an arithmethic progression.


- An annual raise of 5% of the current salary, means that the salary will increase by a constant factor of 1.05, driving to this sequence:

60,000 * 1.05 = 63,000
63,000 * 1.05 = 66,150
66,150 * 1.05 = 69,457.50
69,457.50 * 1.05 = 72,930.375
72,930.375 * 1.05 = 76,576.89
...

In this case, the increase is geometrical because you have that two adjacent terms differentiate by a constant factor, e.g.: 69,457.50 / 66,150 = 1.50.

2)Model each of Mr. Nicholson’s salary options with a recursive sequence that includes his potential earnings for the first three years of employment.
According to the first three terms of each sequence, can you conclude that there is a significant difference in Mr. Nicholson’s potential earnings with each increase option? Use complete sentences to explain your conclusion.

Models

- Atrihmetic progression option

Annual salary the year n= Sn
Initial Salary = S1 = 60,000
difference, d =  3500
number of year: n

Model: S = S1 + (n-1)*d

S = 60,000 + (n-1)*3500

Potential earnings for first three years:

You can use the fomula for the sum of n terms in an arithmetic progression: [S1 + S3]*(n) / (2)

Sum = [60,000 + 67000] * 3 / 2 = 190,500

This is the same that [60,000 + 63,500 + 67,000] = 190,500.

- Geometric progression:

S1 = 60,000
r = 1.05

Sn = S1 * r^(n-1) = 60,000 - (1.05)^(n-1)

Potential earnings the first three years:

60,000 + 63,000 + 66,150 = 189,150

Now you got that there is a substantial difference  in potential earnings with each option: the constant increase of $3500 (arithmetic progression) during three years results in a bigger earning for that time, because 5% of difference the second year is only 3000, and the third year 3150; both below $3500. This results in that the arithmetic progression is better for Mr. Nicholson during the first three years.