Alex earns a [tex]$35,000$[/tex] 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 [tex]$n$[/tex] years of his career?
Choose 1 answer:
A. [tex]$\frac{35,000\left(1-1.03^n\right)}{-0.03}$[/tex]
B. [tex]$\frac{35,000\left(1-0.97^n\right)}{0.03}$[/tex]
C. [tex]$1.03\left(1-35,000^n\right)$[/tex]
Answer (2)
The salaries form a geometric sequence with first term a = 35 , 000 and common ratio r = 1.03 .
The sum of the first n terms of a geometric series is given by S n = a × 1 − r 1 − r n .
Substituting the values of a and r , the total amount earned in n years is 35 , 000 × 1 − 1.03 1 − 1.0 3 n .
Simplifying the expression, the total amount earned is − 0.03 35 , 000 ( 1 − 1.0 3 n ) .
Explanation
Problem Analysis Let's analyze the problem. Alex's salary increases by 3% each year, so his salaries form a geometric sequence. We need to find the sum of this geometric sequence for the first n years.
Salaries Over the Years The salary in the first year is $35,000. The salary in the second year is 35 , 000 \t × 1.03 . The salary in the third year is 35 , 000 × 1.0 3 2 , and so on. The salary in the n -th year is 35 , 000 × 1.0 3 n − 1 . The total amount earned in n years is the sum of this geometric series: 35 , 000 + 35 , 000 \t × 1.03 + 35 , 000 × 1.0 3 2 + ... + 35 , 000 × 1.0 3 n − 1 .
Sum of Geometric Series The sum of a geometric series is given by S n = a × 1 − r 1 − r n , where a is the first term and r is the common ratio. In this case, a = 35 , 000 and r = 1.03 . Therefore, the total amount earned in n years is 35 , 000 × 1 − 1.03 1 − 1.0 3 n = 35 , 000 × − 0.03 1 − 1.0 3 n .
Final Answer The expression that gives the total amount Alex has earned in his first n years of his career is − 0.03 35 , 000 ( 1 − 1.0 3 n ) . This corresponds to option (A).
Examples
Understanding geometric sequences is useful in many financial calculations, such as calculating the future value of an investment with regular percentage growth. For example, if you invest $1000 each year with a 5% annual return, you can use the sum of a geometric series to determine the total value of your investment after a certain number of years. This helps in planning for long-term financial goals like retirement or education.
The total amount Alex earns in his first n years is represented by the expression − 0.03 35 , 000 ( 1 − 1.0 3 n ) , which corresponds to option (A). This formula comes from the sum of a geometric series where his salary increases by 3% each year.
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