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. 5

Explanation

To find the first term of the arithmetic progression (AP), we can use the information given about the sums of the first 10 and 20 terms.

### Step-by-Step Solution:

1. **Understanding the formula for the sum of the first n terms of an AP**:
The sum of the first n terms (S_n) of an arithmetic progression can be expressed as:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
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.

2. **Setting up the equations**:
We know:
- \( S_{10} = 120 \)
- \( S_{20} = 440 \)

Using the formula for \( S_n \):
\[ S_{10} = \frac{10}{2} \times (2a + (10-1)d) = 120 \]
Simplifying this gives:
\[ 5(2a + 9d) = 120 \]
Dividing both sides by 5:
\[ 2a + 9d = 24 \quad \text{(Equation 1)} \]

Now for \( S_{20} \):
\[ S_{20} = \frac{20}{2} \times (2a + (20-1)d) = 440 \]
Simplifying this gives:
\[ 10(2a + 19d) = 440 \]
Dividing both sides by 10:
\[ 2a + 19d = 44 \quad \text{(Equation 2)} \]

3. **Solving the equations**:
We now have a system of two equations:
\[ \text{Equation 1: } 2a + 9d = 24 \]
\[ \text{Equation 2: } 2a + 19d = 44 \]

To eliminate \( 2a \), we can subtract 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. **Finding the first term \( a \)**:
Now that we have \( d \), we can substitute it back into Equation 1 to find \( a \):
\[ 2a + 9(2) = 24 \]
Simplifying this gives:
\[ 2a + 18 = 24 \]
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:
\[ \boxed{3} \]