The sum of the first 9 terms of an AP is 81 and that of its first 20 terms is 400. Find the first term and the common difference of the AP

Explanation

To solve the problem step by step, we will use the formula for the sum of the first \( n \) terms of an arithmetic progression (AP):

\[ S_n = \frac{n}{2} \left(2A + (n-1)D\right) \]

where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( A \) is the first term,
- \( D \) is the common difference,
- \( n \) is the number of terms.

### Step 1: Set up the equations

Given:
- The sum of the first 9 terms \( S_9 = 81 \)
- The sum of the first 20 terms \( S_{20} = 400 \)

Using the sum formula for the first 9 terms:

\[ S_9 = \frac{9}{2} \left(2A + 8D\right) = 81 \]

Multiplying both sides by 2 to eliminate the fraction:

\[ 9(2A + 8D) = 162 \]

Dividing both sides by 9:

\[ 2A + 8D = 18 \quad \text{(Equation 1)} \]

Now, using the sum formula for the first 20 terms:

\[ S_{20} = \frac{20}{2} \left(2A + 19D\right) = 400 \]

Multiplying both sides by 2:

\[ 20(2A + 19D) = 800 \]

Dividing both sides by 20:

\[ 2A + 19D = 40 \quad \text{(Equation 2)} \]

### Step 2: Solve the equations

Now we have two equations:

1. \( 2A + 8D = 18 \) (Equation 1)
2. \( 2A + 19D = 40 \) (Equation 2)

Next, we will subtract Equation 1 from Equation 2:

\[ (2A + 19D) - (2A + 8D) = 40 - 18 \]

This simplifies to:

\[ 11D = 22 \]

Dividing both sides by 11:

\[ D = 2 \]

### Step 3: Find the first term \( A \)

Now that we have \( D \), we can substitute it back into either Equation 1 or Equation 2 to find \( A \). We'll use Equation 1:

\[ 2A + 8D = 18 \]

Substituting \( D = 2 \):

\[ 2A + 8(2) = 18 \]

This simplifies to:

\[ 2A + 16 = 18 \]

Subtracting 16 from both sides:

\[ 2A = 2 \]

Dividing both sides by 2:

\[ A = 1 \]

### Final Answer

The first term \( A \) is 1 and the common difference \( D \) is 2.

### Summary of the Solution
- First term \( A = 1 \)
- Common difference \( D = 2 \)