Mr. nicholson accepts a job that pays an annual salary of $60,000. in his employment contract, he is given the option of choosing a) an annual raise of $3,500 or b) an annual raise of 5% of his current salary.

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Mr. nicholson accepts a job that pays an annual salary of $60,000. in his employment contract, he is given the option of choosing a) an annual raise of $3,500 or b) an annual raise of 5% of his current salary.

So do we say which one is better or worse?

Step-by-step explanation:

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.

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He should choose a) an annual raise of $3500.

This is because if you take 5% of $60,000 its only $3,000

So do we say which one is better or worse?

I believe that the answer is b since it’s an annual raise

He should choose a) an annual raise of $3500.

This is because if you take 5% of $60,000 its only $3,000

See below

Step-by-step explanation:

1. Arithmetic: Add 3,500

The annual raise of 3,500 means add 3,500 to the previous year's salary

60,000 + 3,500 = 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

Geometric: Multiply by 1.05

The annual raise by 5% of his current salary means multiply by 1.05.

60,000 x 1.05 = 63,000

63,000 x 1.05 = 66,150

66,150 x 1.05 = 69,457.50

69,457.50 x 1.05 = 72,930.38 rounded

72,930.375 x 1.05 = 76,576.89

The first sequence is arithmetic because we add the same number (3,500) to the preceding term. The second sequence is geometric because we multiply the preceding term by the same number always (1.05.)

2a. Arithmetic - New salary is $3,500 greater each year than last year's salary

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

Geometric - New salary is 5% more each year than last year's salary

60,000 + (1.05)^(n-1)

2b. Arithmetic Earnings over 3 years

60,000 + 63,500 + 67,000 = 190,500

Geometric Earnings over 3 years

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

There is a 1,000 dollar difference. In this case, the arithmetic increase of 3,500 dollars would be better for Mr. Nicholson. 1,000 dollars may or may not be considered a big difference. In my opinion, I'd say there is a slight difference between the two

3.Arithmetic

a(9) = 3,500 + a(9-1)

a(9) = 3,500 + 89,000

a(9) = 92,500

Geometric

a(9) = 1.05 x a(9-1)

a(9) = 1.05 x 84,425.90

a(9) = 88,647.20

4. In this case, both the 3 year and 9 year time frames favor the arithmetic increase of $3,500. At 3 years, he would have 190,500 compared to the geometric salary of 189,150. However, this is a small difference. If he is going to be at the company for 9 years, then definitely he should choose the first opportunity. 92,500 is significantly more money than 88,647.20. So, longer time frames only make the first opportunity, which is better to begin with, shine even more.

I'm always happy to help 🙂