A manufacturer of TV sets produces 600 units in the third year and 700 units in the 7th year. Assuming that the production increases uniformly by a fixed number every year, find:
(i) the production in the first year.
(ii) the production in the 10th year.
(iii) the total production in 7 years.

Answer

To solve the problem step by step, we will assume the production in the first year is \( a \) and the common difference (the uniform increase in production every year) is \( d \).

### Step 1: Set up the equations based on the information given.

- The production in the 3rd year (T3) is given as 600 units.
- The production in the 7th year (T7) is given as 700 units.

Using the formula for the nth term of an arithmetic progression (AP), we can express these as:

1. For the 3rd year:
\[ T_3 = a + 2d = 600 \quad \text{(Equation 1)} \]

2. For the 7th year:
\[ T_7 = a + 6d = 700 \quad \text{(Equation 2)} \]

### Step 2: Subtract Equation 1 from Equation 2.

To eliminate \( a \), we subtract Equation 1 from Equation 2:

\[ (a + 6d) - (a + 2d) = 700 - 600 \]

This simplifies to:

\[ 4d = 100 \]

### Step 3: Solve for \( d \).

Now, we can solve for \( d \):

\[ d = \frac{100}{4} = 25 \]

### Step 4: Substitute \( d \) back into one of the equations to find \( a \).

We can substitute \( d \) back into Equation 1 to find \( a \):

\[ a + 2(25) = 600 \]

This simplifies to:

\[ a + 50 = 600 \]

Now, solving for \( a \):

\[ a = 600 - 50 = 550 \]

### Step 5: Find the production in the 10th year (T10).

Using the formula for the 10th term:

\[ T_{10} = a + 9d \]

Substituting the values of \( a \) and \( d \):

\[ T_{10} = 550 + 9(25) = 550 + 225 = 775 \]

### Step 6: Calculate the total production in 7 years (S7).

The sum of the first \( n \) terms of an AP is given by:

\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]

For the first 7 years:

\[ S_7 = \frac{7}{2} \times (2(550) + (7-1)(25)) \]

Calculating this step-by-step:

1. Calculate \( 2a \):
\[ 2(550) = 1100 \]

2. Calculate \( (n-1)d \):
\[ (7-1)(25) = 6 \times 25 = 150 \]

3. Now substitute back:
\[ S_7 = \frac{7}{2} \times (1100 + 150) = \frac{7}{2} \times 1250 \]

4. Calculate:
\[ S_7 = \frac{7 \times 1250}{2} = \frac{8750}{2} = 4375 \]

### Final Answers:
(i) The production in the first year is **550 units**.
(ii) The production in the 10th year is **775 units**.
(iii) The total production in 7 years is **4375 units**.