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. a) Model each of Mr. Nicholson’s salary options with a recursive sequence that includes his potential earnings for the first three years of employment.
b) 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.

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.

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