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Let's break down the problem and solve it step by step:

### Given:

- The company made a profit of **$250,000** in the year **2000**.
- Two plans for increasing profits are considered:
  - **Plan A:** The annual profit increases each year by **5%** of its value in the previous year.
  - **Plan B:** The annual profit increases by a constant amount **$D** each year.
  
We are tasked with:

1. **For Plan A:**
   - Find the profit for the year **2008**.
   - Find the total profit for the **10 years** (from 2000 to 2009 inclusive).

2. **For Plan B:**
   - Find the value of **$D** such that the total profit for the 10 years under Plan B is the same as the total profit for the 10 years under Plan A.

---

### Part 1: **Plan A**

#### (i) Finding the profit for the year **2008** under Plan A:
The profit grows by **5%** each year, so this is a **geometric sequence**.

- The initial profit in **2000** is **$250,000**.
- The formula for the profit in year $ n $ (where $ n $ starts at 0 for 2000) is:
  \[
  P_n = 250,000 \times (1.05)^n
  \]
  
For the year **2008**, $ n = 8 $ (since 2008 is 8 years after 2000):
\[
P_8 = 250,000 \times (1.05)^8
\]

Let's calculate that.

#### (ii) Finding the total profit for the 10 years (2000 to 2009 inclusive) under Plan A:
The total profit for the 10 years is the sum of a geometric series:
\[
S_{10} = 250,000 \times \frac{(1.05^{10} - 1)}{0.05}
\]
This gives the total profit from 2000 to 2009.

### Part 2: **Plan B**

#### (iii) Find the value of **$D** such that the total profit under Plan B is the same as under Plan A.

Under Plan B, the profit increases by a constant amount **$D** each year, which forms an **arithmetic sequence**.

- The profit in 2000 is **$250,000**.
- The formula for the profit in year $ n $ under Plan B is:
  \[
  P_n = 250,000 + n \times D
  \]
  
The total profit for the 10 years (2000 to 2009) under Plan B is the sum of an arithmetic series:
\[
S_{10} = \frac{10}{2} \times (2 \times 250,000 + 9 \times D)
\]
We can equate this to the total profit for Plan A and solve for **$D**.

---

I'll start by calculating the results for Plan A first.### Plan A:

1. **The profit for the year 2008** is **$369,364** (rounded to 3 significant figures).
2. **The total profit for the 10 years (2000 to 2009 inclusive)** is **$3,144,473** (rounded to 3 significant figures).

Next, I'll solve for the constant increase **$D** in **Plan B** by equating the total profit for the 10 years under both plans. Let's proceed with that calculation.### Plan B:

The constant increase **$D** for which the total profit over the 10 years (2000 to 2009) would be the same as in Plan A is approximately **$14,322** (rounded to 3 significant figures).

Would you like further clarifications or more details? Here are some related questions:

1. How would the profits in 2005 compare under both plans?
2. How can we adjust the growth rate in Plan A to match a specific target?
3. What if the initial profit was higher in 2000? How would that impact the results?
4. How would inflation adjustments affect the total profits in either plan?
5. Can we compute the break-even point between Plan A and Plan B over a different period?

**Tip:** For geometric sequences, each term is multiplied by a constant factor, whereas in arithmetic sequences, a constant amount is added to each term. Understanding this difference is key to solving similar problems.

A small trading company made a profit of $250,000 in the year 2000. The company considered two different plans, Plan A and Plan B, for increasing its profits. Under Plan A, the annual profit would increase each year by 5% of its value in the preceding year. Find for Plan A, (i) the profit for the year 2008, (ii) the total profit for the 10 years from 2000 to 2009 inclusive. Under Plan B, the annual profit would increase each year by a constant amount D. (iii) Find the value of D for which the total profit for the 10 years from 2000 to 2009 inclusive would be the same for both plans.

Solve the problem of a trading company's profit growth with two plans: Plan A (geometric sequence growth) and Plan B (arithmetic sequence growth). Find the profit for 2008 under Plan A, the total profit from 2000 to 2009, and the constant D in Plan B that equates both plans' profits. Suitable for students in Grades 10-12.

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