The problem provided discusses the production of cigarettes by a factory. Let's break down and solve it step by step.

Given Data:

1. Initial production in year 1: 1,000,000 packs.
2. Production in year 6: 1,750,000 packs.

We assume the production increases at a constant rate each year.

a. Find the additional production per year:

The increase in production from year 1 to year 6 is linear, meaning the same amount is added every year. To find the annual increase, we can use the formula for the slope between two points in arithmetic sequences:

$\text{Increase per year} = \frac{\text{Production in year 6} - \text{Production in year 1}}{6 - 1}$

Let's calculate this.

$\text{Increase per year} = \frac{1,750,000 - 1,000,000}{6 - 1} = \frac{750,000}{5} = 150,000$

So, the production increases by 150,000 packs per year.

b. Find the production in year 11:

Since the production increases by 150,000 packs per year, we can calculate the production in year 11 by extending the arithmetic sequence.

The formula for the $n$-th term of an arithmetic sequence is:

$P_n = P_1 + (n - 1) \times d$

Where:

• $P_n$ is the production in year $n$,
• $P_1$ is the production in year 1 (1,000,000 packs),
• $d$ is the increase per year (150,000),
• $n = 11$.

Substituting the values:

$P_{11} = 1,000,000 + (11 - 1) \times 150,000$ $P_{11} = 1,000,000 + 10 \times 150,000$ $P_{11} = 1,000,000 + 1,500,000 = 2,500,000$

Thus, the production in year 11 will be 2,500,000 packs.

Summary:

• The additional production per year is 150,000 packs.
• The production in year 11 will be 2,500,000 packs.

Would you like more details on any step or further clarification? Here are some related questions for you:

1. What if the production increased by a different amount every year instead of a constant rate?
2. How would we calculate the production in year 20?
3. Can we model this problem using a different method like geometric sequences?
4. What if the production in year 1 had been lower—how would that change the result?
5. Can this method apply to other real-world problems, such as factory output for different products?

Tip: Always verify if the given pattern (constant increase) holds in similar real-world scenarios before making predictions.