A manufacturer of cricket bat produced 2000 units in `2^(nd)` year and 5000 units in the `5^(th)` year. Assuming that production increases uniformly by a fixed number every year, find the total production in 20 years.

Options

A. 125000 units
B. 180000 units
C. 60000 units
D. 210000 units

Answer

To solve the problem step by step, we will first establish the relationship between the years and the production of cricket bats.

### Step 1: Define the variables
Let:
- \( A \) = the production in the first year
- \( D \) = the fixed increase in production each year

### Step 2: Write the production equations
From the problem, we know:
- In the 2nd year (T2), the production is \( A + D = 2000 \) (1)
- In the 5th year (T5), the production is \( A + 4D = 5000 \) (2)

### Step 3: Set up the equations
From equation (1):
\[ A + D = 2000 \]
From equation (2):
\[ A + 4D = 5000 \]

### Step 4: Solve the equations
Subtract equation (1) from equation (2):
\[ (A + 4D) - (A + D) = 5000 - 2000 \]
This simplifies to:
\[ 3D = 3000 \]
Thus,
\[ D = 1000 \]

### Step 5: Substitute D back to find A
Now substitute \( D \) back into equation (1):
\[ A + 1000 = 2000 \]
So,
\[ A = 1000 \]

### Step 6: Write the production formula for any year
The production in the \( n^{th} \) year can be expressed as:
\[ T_n = A + (n-1)D \]
Substituting \( A \) and \( D \):
\[ T_n = 1000 + (n-1) \times 1000 = 1000n \]

### Step 7: Calculate total production in 20 years
The total production over 20 years (S20) is the sum of the productions from year 1 to year 20:
\[ S_{20} = T_1 + T_2 + T_3 + \ldots + T_{20} \]
Using the formula for \( T_n \):
\[ S_{20} = 1000 \times 1 + 1000 \times 2 + 1000 \times 3 + \ldots + 1000 \times 20 \]
This can be factored as:
\[ S_{20} = 1000 \times (1 + 2 + 3 + \ldots + 20) \]

### Step 8: Use the formula for the sum of the first n natural numbers
The sum of the first \( n \) natural numbers is given by:
\[ \frac{n(n + 1)}{2} \]
For \( n = 20 \):
\[ \text{Sum} = \frac{20 \times 21}{2} = 210 \]

### Step 9: Calculate S20
Now substituting back:
\[ S_{20} = 1000 \times 210 = 210000 \]

### Final Answer
The total production in 20 years is **210000 units**.

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