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Harmony earns a 
$42,000 salary in the first year of her career. Each year, she gets a 
4% raise.
Which expression gives the total amount Harmony has earned in her first 
n years of her career?
Choose 1 answer:
(A) 
42,000((1-0.04^(n))/(0.96))
(B) 
42,000((1-1.04^(n))/(-0.04))
(c) 
42,000((1-1.04^(n))/(0.96))
(D) 
42,000((1-0.96^(n))/(-0.04))

Harmony earns a $42,000 \$ 42,000 salary in the first year of her career. Each year, she gets a 4% 4 \% raise.\newlineWhich expression gives the total amount Harmony has earned in her first n n years of her career?\newlineChoose 11 answer:\newline(A) 42,000(10.04n0.96) 42,000\left(\frac{1-0.04^{n}}{0.96}\right) \newline(B) 42,000(11.04n0.04) 42,000\left(\frac{1-1.04^{n}}{-0.04}\right) \newline(C) 42,000(11.04n0.96) 42,000\left(\frac{1-1.04^{n}}{0.96}\right) \newline(D) 42,000(10.96n0.04) 42,000\left(\frac{1-0.96^{n}}{-0.04}\right)

Full solution

Q. Harmony earns a $42,000 \$ 42,000 salary in the first year of her career. Each year, she gets a 4% 4 \% raise.\newlineWhich expression gives the total amount Harmony has earned in her first n n years of her career?\newlineChoose 11 answer:\newline(A) 42,000(10.04n0.96) 42,000\left(\frac{1-0.04^{n}}{0.96}\right) \newline(B) 42,000(11.04n0.04) 42,000\left(\frac{1-1.04^{n}}{-0.04}\right) \newline(C) 42,000(11.04n0.96) 42,000\left(\frac{1-1.04^{n}}{0.96}\right) \newline(D) 42,000(10.96n0.04) 42,000\left(\frac{1-0.96^{n}}{-0.04}\right)
  1. Identify Geometric Series: We need to find the sum of a geometric series because Harmony's salary increases by a constant percentage each year. The first term of the series a1a_1 is her initial salary, which is $42,000\$42,000. The common ratio rr is 1+0.041 + 0.04, because she gets a 44\% raise each year. The formula for the sum of the first nn terms of a geometric series is Sn=a1×(1rn)/(1r)S_n = a_1 \times (1 - r^n) / (1 - r) for r1r \neq 1.
  2. Plug in Values: Let's plug in the values we have into the formula. The initial salary a1a_1 is $42,000\$42,000, and the common ratio rr is 1.041.04 (since the raise is 4%4\%, or 0.040.04 as a decimal). The formula becomes Sn=42,000×(11.04n)/(11.04)S_n = 42,000 \times (1 - 1.04^n) / (1 - 1.04).
  3. Simplify Denominator: Simplify the denominator of the formula. We have 11.041 - 1.04, which is 0.04-0.04. So, the formula now is Sn=42,000×(11.04n)/(0.04)S_n = 42,000 \times (1 - 1.04^n) / (-0.04).
  4. Match with Choices: We can now match our formula with the given choices. The correct expression must have the initial salary multiplied by the sum of the geometric series, which we have found to be Sn=42,000×(11.04n)/(0.04)S_n = 42,000 \times (1 - 1.04^n) / (-0.04). This matches with choice (B).

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