Answer:
Option A
Step-by-step explanation:
Option A is an arithmetic sequence.
Each week, the salary goes up by a fixed $50.
To verify this, subtract any two consecutive weeks' salaries.
For example: $250 - $200 = $50; $350 - $300 = $50, etc.
The common difference in 50.
We have an arithmetic sequence with 20 terms. The first term is $200. We need to find the sum of the 20 terms.
The sum of an arithmetic sequence is given by the formula:
[tex] S_n = \dfrac{n}{2} \times [2a_1 + (n - 1)d] [/tex]
S_n = sum of first n terms
n = number of terms = 20
a_1 = first term = 200
d = common difference = 50
[tex] S_{20} = \dfrac{20}{2} \times [2(200) + (20 - 1)(50)] [/tex]
[tex] S_{20} = 10 \times [400 + 19(50)] [/tex]
[tex] S_{20} = 10 \times [400 + 19(50)] [/tex]
[tex] S_{20} = 13500 [/tex]
Option A gives a total of $13,500 for the first 20 weeks.
Option B is a geometric sequence in which the salary goes up by 10% each week. To verify this, divide any salary by the previous week's salary.
For example: $220/$200 = 1.10; $266.20/$242 = 1.10; in each case, each salary is 1.1 times the previous week's salary which means a 10% increase. The common ratio of the geometric sequence is 1.1.
We need the formula for the sum of the first n terms of a geometric sequence.
[tex] S_n = \dfrac{a_1(1 - r^n)}{1 - r} [/tex]
[tex] S_{20} = \dfrac{200(1 - 1.1^{20})}{1 - 1.1} [/tex]
[tex] S_{20} = \dfrac{200(1 - 6.7274999)}{-0.1} [/tex]
[tex] S_{20} = 11455 [/tex]
Option B gives a total of $11,455 for the first 20 weeks.
Answer: Option A
