Alex earns a $35,000 salary in the first year of his career. Each year, he gets a 3% raise.Which expression gives the total amount Alex has earned in his first n years of his career?Choose 1 answer:(A) −0.0335,000(1−1.03n)(B) 0.0335,000(1−0.97n)(C) 35,0001.03(1−35,000n)(D) 35,0000.97(1−35,000n)
Q. Alex earns a $35,000 salary in the first year of his career. Each year, he gets a 3% raise.Which expression gives the total amount Alex has earned in his first n years of his career?Choose 1 answer:(A) −0.0335,000(1−1.03n)(B) 0.0335,000(1−0.97n)(C) 35,0001.03(1−35,000n)(D) 35,0000.97(1−35,000n)
Problem Understanding: Understand the problem.Alex earns a salary that increases by 3% each year. We need to find the total amount he has earned over n years. This is a geometric series because each term increases by a constant percentage (3%) from the previous one.
Geometric Series Formula: Identify the formula for the sum of a geometric series.The sum S of the first n terms of a geometric series with the first term a1 and common ratio r is given by:S=a1⋅1−r1−rn, if r=1In this problem, a1 is the initial salary ($35,000), and r is 1.03 (since the salary increases by 3% each year).
Substituting Values: Substitute the values into the formula.Using the formula for the sum of a geometric series:S=35000×(1−1.03n)/(1−1.03)We simplify the denominator to −0.03 because 1−1.03=−0.03.
Choosing the Correct Expression: Choose the correct expression.We need to find the expression that matches our derived formula. The correct expression must have 35000 as the initial salary, 1.03 as the common ratio, and −0.03 in the denominator to represent the 3% increase.
Matching the Expression to Choices: Match the expression to the choices.Looking at the choices, we can eliminate (C) and (D) because they do not have the correct structure of the geometric series sum formula. Between (A) and (B), (A) has the correct structure and signs according to our formula:(35000×(1−1.03n))/(−0.03)
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