Questions: Select all that apply A company's revenue has been increasing according to an arithmetic gradient for the past seven years. The revenue in year 1 was 4,000,000 and it increased by 1,000,000 each year thereafter. The revenue in year 4 is A dollars and the present worth of the cash flow series in year 0 is B dollars. Assuming an interest rate of 9% per year, determine the values of A and B.

Select all that apply A company's revenue has been increasing according to an arithmetic gradient for the past seven years. The revenue in year 1 was 4,000,000 and it increased by 1,000,000 each year thereafter. The revenue in year 4 is A dollars and the present worth of the cash flow series in year 0 is B dollars. Assuming an interest rate of 9% per year, determine the values of A and B.

Transcript text: Select all that apply A company's revenue has been increasing according to an arithmetic gradient for the past seven years. The revenue in year 1 was $\$ 4,000,000$ and it increased by $\$ 1,000,000$ each year thereafter. The revenue in year 4 is A dollars and the present worth of the cash flow series in year 0 is $B$ dollars. Assuming an interest rate of $9 \%$ per year, determine the values of $A$ and $B$.

Solution

Solution Steps

To solve this problem, we need to determine two values: A and B.

Finding A (Revenue in Year 4): Since the revenue increases by a constant amount each year, we can use the formula for the nth term of an arithmetic sequence. The revenue in year 4 can be calculated by adding the initial revenue to three times the annual increase.
Finding B (Present Worth of Cash Flow Series): The present worth of the cash flow series can be calculated using the formula for the present value of an arithmetic gradient series. We need to discount each year's revenue back to year 0 using the given interest rate.

Step 1: Calculate Revenue in Year 4

The revenue follows an arithmetic sequence with an initial revenue of \$4,000,000 and an annual increase of \$1,000,000. The revenue in year 4 can be calculated using the formula for the nth term of an arithmetic sequence:

\[ A = \text{initial\_revenue} + 3 \times \text{annual\_increase} \]

Substituting the given values:

\[ A = 4,000,000 + 3 \times 1,000,000 = 7,000,000 \]

Step 2: Calculate Present Worth of Cash Flow Series

The present worth $ B $ of the cash flow series is calculated by discounting each year's revenue back to year 0 using the interest rate of 9%. The formula for the present value of each year's revenue is:

\[ \text{Present\_Value} = \frac{\text{Revenue in Year } n}{(1 + \text{interest\_rate})^n} \]

The total present worth is the sum of the present values of revenues from year 1 to year 7:

\[ B = \sum_{n=1}^{7} \frac{\text{Revenue in Year } n}{(1 + 0.09)^n} \]

Calculating this sum gives:

\[ B \approx 33,506,401.5755 \]