Suppose a business earned a profit of $8,000 in its first month of operation.  Unfortunately, its profit decreased by 2% each month for the next year.  Find the total profit for the first year.


The problem describes a scenario where the business initially earns a profit of $8,000, and the profit decreases by 2% each month over the next year. To find the total profit for the first year, we can model this as a geometric series.

### Step-by-Step Solution:

1. **Initial Profit:**
   - The business starts with an $8,000 profit in the first month.

2. **Monthly Decrease:**
   - The profit decreases by 2% each month. Hence, each month's profit is 98% (or 0.98 times) of the previous month’s profit.

3. **Geometric Series Setup:**
   - The first term of the series ($a_1$) is $8,000.
   - The common ratio ($r$) is 0.98.
   - We want to sum the profits for 12 months.

4. **Formula for the sum of a geometric series:**
   The sum $S_n$ of the first $n$ terms of a geometric series is given by the formula:
   \[
   S_n = a_1 \frac{1 - r^n}{1 - r}
   \]
   where:
   - $a_1 = 8,000$ (initial profit),
   - $r = 0.98$ (common ratio),
   - $n = 12$ (number of months).

5. **Substitute the values:**
   \[
   S_{12} = 8000 \times \frac{1 - (0.98)^{12}}{1 - 0.98}
   \]
   Let's calculate this.

The total profit for the first year is approximately **$86,113.31**.

Would you like more details or have any questions?

Here are five related questions to expand your understanding:
1. How would the total profit change if the decrease rate was 5% instead of 2%?
2. What would be the total profit if the decrease rate remained the same but for a two-year period?
3. Can you calculate the profit for each individual month in this scenario?
4. How would the total profit differ if the decrease happened every two months instead of monthly?
5. What if the profit decreased by a fixed dollar amount instead of a percentage?

**Tip:** Geometric series are useful in finance and business for modeling situations where there is exponential growth or decay, such as interest rates or depreciation.

Suppose a business earned a profit of $8,000 in its first month of operation. Unfortunately, its profit decreased by 2% each month for the next year. Find the total profit for the first year.

Discover how to calculate the total profit for a business that starts with $8,000 in the first month and experiences a 2% monthly decrease. This solution uses the geometric series formula to find the total profit for the first year, approximately $86,113.31. Suitable for Grades 10-12 students learning algebra and exponential decay.

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