Question

What is the company’s production in the first year ?

Answer 1

The production forms an arithmetic progression (AP) where:
\[a = \text{Production in the first year}, \quad d = \text{Increase in production every year}.\]
The production in the $n$-th year is given by:
\[a_n = a + (n-1)d.\]
For the 6th year:
\[800 = a + (6 - 1)d \implies 800 = a + 5d. \tag{1}\]
For the 9th year:
\[1130 = a + (9 - 1)d \implies 1130 = a + 8d. \tag{2}\]
Subtract (1) from (2):
\[1130 - 800 = a + 8d - (a + 5d) \implies 330 = 3d \implies d = 110.\]
Substitute $d = 110$  into (1):
\[800 = a + 5(110) \implies 800 = a + 550 \implies a = 250.\]
Correct Answer: Production in the first year = 250 rollers.

Answer 2

Step 1: Understanding the problem:
We are told that the company increases its production uniformly by a fixed number each year. We are given the following information: 
- The company produces 800 rollers in the 6th year. 
- The company produces 1130 rollers in the 9th year.
We need to find the increase in the company’s production every year, which is the common difference in an arithmetic progression (A.P.), where the production increases by a fixed number each year.

Step 2: Defining the terms:
Let the number of rollers produced in the first year be \(a\), and let the increase in production every year be \(d\) (the common difference in the A.P.). The production in the \(n\)-th year can be given by the formula:
\[ a_n = a + (n-1) \cdot d \] where \(a_n\) is the production in the \(n\)-th year.

Step 3: Using the given information:
We know: - In the 6th year, the production is 800 rollers, i.e., \( a_6 = 800 \), - In the 9th year, the production is 1130 rollers, i.e., \( a_9 = 1130 \). Using the formula for the \(n\)-th term, we have the following two equations: 1. For the 6th year: \[ a_6 = a + (6-1) \cdot d = a + 5d = 800 \] \[ a + 5d = 800 \quad \text{(Equation 1)} \] 2. For the 9th year: \[ a_9 = a + (9-1) \cdot d = a + 8d = 1130 \] \[ a + 8d = 1130 \quad \text{(Equation 2)} \]

Step 4: Solving the system of equations:
We have the system of equations: 1. \( a + 5d = 800 \) 2. \( a + 8d = 1130 \) Subtract Equation 1 from Equation 2 to eliminate \(a\):
\[ (a + 8d) - (a + 5d) = 1130 - 800 \] \[ 3d = 330 \] \[ d = \frac{330}{3} = 110 \]

Step 5: Conclusion:
The increase in the company’s production every year is 110 rollers.

Answer 3

Step 1: Understanding the problem:
We need to find:
1. The company’s production in the 8th year.
2. The total production in the first 6 years.

Step 2: Given information:
From the previous solution, we know that the company’s production increases uniformly by 110 rollers each year (i.e., \( d = 110 \)).
Also, the production in the 6th year is 800 rollers, which gives us the following information for the arithmetic progression (A.P.):
- \( a_6 = a + 5d = 800 \)
- \( d = 110 \)

Step 3: Finding the production in the 8th year:
The production in the 8th year, \( a_8 \), can be calculated using the formula for the nth term of an arithmetic progression:
\[ a_n = a + (n-1) \cdot d \]
For the 8th year, \( n = 8 \):
\[ a_8 = a + (8-1) \cdot d = a + 7d \]
Substitute the known values of \( a \) and \( d \):
\[ a_6 = a + 5d = 800 \quad \Rightarrow \quad a + 5 \times 110 = 800 \]
\[ a + 550 = 800 \]
\[ a = 800 - 550 = 250 \]
Now, substitute \( a = 250 \) and \( d = 110 \) into the formula for \( a_8 \):
\[ a_8 = 250 + 7 \times 110 = 250 + 770 = 1020 \]
Thus, the production in the 8th year is 1020 rollers.

Step 4: Finding the total production in the first 6 years:
The total production in the first 6 years is the sum of the productions in the 1st, 2nd, 3rd, 4th, 5th, and 6th years. The sum of the first \(n\) terms of an arithmetic progression is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1) \cdot d) \]
For the first 6 years (\(n = 6\)):
\[ S_6 = \frac{6}{2} \times (2 \times 250 + (6-1) \cdot 110) \]
Simplify the equation:
\[ S_6 = 3 \times (500 + 5 \times 110) = 3 \times (500 + 550) = 3 \times 1050 = 3150 \]
Thus, the total production in the first 6 years is 3150 rollers.

Step 5: Conclusion:
1. The company’s production in the 8th year is 1020 rollers.
2. The company’s total production in the first 6 years is 3150 rollers.