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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) 
(35,000(1-1.03^(n)))/(-0.03)
(B) 
(35,000(1-0.97^(n)))/(0.03)
(c) 
(1.03(1-35,000^(n)))/(35,000)
(D) 
(0.97(1-35,000^(n)))/(35,000)

Alex earns a $35,000 \$ 35,000 salary in the first year of his career. Each year, he gets a 3% 3 \% raise.\newlineWhich expression gives the total amount Alex has earned in his first n n years of his career?\newlineChoose 11 answer:\newline(A) 35,000(11.03n)0.03 \frac{35,000\left(1-1.03^{n}\right)}{-0.03} \newline(B) 35,000(10.97n)0.03 \frac{35,000\left(1-0.97^{n}\right)}{0.03} \newline(C) 1.03(135,000n)35,000 \frac{1.03\left(1-35,000^{n}\right)}{35,000} \newline(D) 0.97(135,000n)35,000 \frac{0.97\left(1-35,000^{n}\right)}{35,000}

Full solution

Q. Alex earns a $35,000 \$ 35,000 salary in the first year of his career. Each year, he gets a 3% 3 \% raise.\newlineWhich expression gives the total amount Alex has earned in his first n n years of his career?\newlineChoose 11 answer:\newline(A) 35,000(11.03n)0.03 \frac{35,000\left(1-1.03^{n}\right)}{-0.03} \newline(B) 35,000(10.97n)0.03 \frac{35,000\left(1-0.97^{n}\right)}{0.03} \newline(C) 1.03(135,000n)35,000 \frac{1.03\left(1-35,000^{n}\right)}{35,000} \newline(D) 0.97(135,000n)35,000 \frac{0.97\left(1-35,000^{n}\right)}{35,000}
  1. Problem Understanding: Understand the problem.\newlineAlex earns a salary that increases by 3%3\% each year. We need to find the total amount he has earned over nn years. This is a geometric series because each term increases by a constant percentage (3%3\%) from the previous one.
  2. Geometric Series Formula: Identify the formula for the sum of a geometric series.\newlineThe sum SS of the first nn terms of a geometric series with the first term a1a_1 and common ratio rr is given by:\newlineS=a11rn1rS = a_1 \cdot \frac{1 - r^n}{1 - r}, if r1r \neq 1\newlineIn this problem, a1a_1 is the initial salary ($35,000\$35,000), and rr is 1.031.03 (since the salary increases by 3%3\% each year).
  3. Substituting Values: Substitute the values into the formula.\newlineUsing the formula for the sum of a geometric series:\newlineS=35000×(11.03n)/(11.03)S = 35000 \times (1 - 1.03^n) / (1 - 1.03)\newlineWe simplify the denominator to 0.03-0.03 because 11.03=0.031 - 1.03 = -0.03.
  4. Choosing the Correct Expression: Choose the correct expression.\newlineWe need to find the expression that matches our derived formula. The correct expression must have 3500035000 as the initial salary, 1.031.03 as the common ratio, and 0.03-0.03 in the denominator to represent the 3%3\% increase.
  5. Matching the Expression to Choices: Match the expression to the choices.\newlineLooking 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:\newline(35000×(11.03n))/(0.03)(35000 \times (1 - 1.03^n)) / (-0.03)

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