The sum of first 10 terms and 20 terms of an AP are 120 and 440, respectively.
What is its first term ?

Options

A. 2
B. 3
C. 4
D. 1

Answer

To find the first term of the arithmetic progression (AP), we can use the information provided about the sums of the first 10 and 20 terms. Let's denote the first term as \( A \) and the common difference as \( D \).

### Step-by-Step Solution:

1. **Write the formula for the sum of the first \( n \) terms of an AP:**
The sum \( S_n \) of the first \( n \) terms of an arithmetic progression is given by the formula:
\[ S_n = \frac{n}{2} \times (2A + (n - 1)D) \]

2. **Set up the equations using the given sums:**
For the first 10 terms, we know:
\[ S_{10} = 120 \]
Plugging into the formula:
\[ 120 = \frac{10}{2} \times (2A + 9D) \]
Simplifying this:
\[ 120 = 5 \times (2A + 9D) \]
Dividing both sides by 5:
\[ 2A + 9D = 24 \quad \text{(Equation 1)} \]

For the first 20 terms, we know:
\[ S_{20} = 440 \]
Plugging into the formula:
\[ 440 = \frac{20}{2} \times (2A + 19D) \]
Simplifying this:
\[ 440 = 10 \times (2A + 19D) \]
Dividing both sides by 10:
\[ 2A + 19D = 44 \quad \text{(Equation 2)} \]

3. **Subtract Equation 1 from Equation 2:**
Now, we will eliminate \( 2A \) by subtracting Equation 1 from Equation 2:
\[ (2A + 19D) - (2A + 9D) = 44 - 24 \]
This simplifies to:
\[ 10D = 20 \]
Dividing both sides by 10 gives:
\[ D = 2 \]

4. **Substitute \( D \) back into Equation 1:**
Now that we have \( D \), we can substitute it back into Equation 1 to find \( A \):
\[ 2A + 9(2) = 24 \]
Simplifying this:
\[ 2A + 18 = 24 \]
Subtracting 18 from both sides:
\[ 2A = 6 \]
Dividing both sides by 2 gives:
\[ A = 3 \]

### Conclusion:
The first term \( A \) of the arithmetic progression is \( 3 \).