The sum of first 10 terms and 20 terms of an AP are 120 and 440 respectively.
What is the common difference?

Options

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

Explanation

To find the common difference of the arithmetic progression (AP) given the sum of the first 10 terms and the first 20 terms, we can follow these steps:

### Step 1: Write 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 calculated using the formula:
\[ S_n = \frac{n}{2} \times (2A + (n-1)D) \]
where:
- \( S_n \) = sum of the first n terms
- \( A \) = first term
- \( D \) = common difference
- \( n \) = number of terms

### Step 2: Set up equations for the given sums
We know:
- \( S_{10} = 120 \)
- \( S_{20} = 440 \)

Using the formula for \( S_{10} \):
\[ S_{10} = \frac{10}{2} \times (2A + (10-1)D) = 120 \]
This simplifies to:
\[ 5 \times (2A + 9D) = 120 \]
Dividing both sides by 5:
\[ 2A + 9D = 24 \quad \text{(Equation 1)} \]

Now, using the formula for \( S_{20} \):
\[ S_{20} = \frac{20}{2} \times (2A + (20-1)D) = 440 \]
This simplifies to:
\[ 10 \times (2A + 19D) = 440 \]
Dividing both sides by 10:
\[ 2A + 19D = 44 \quad \text{(Equation 2)} \]

### Step 3: Solve the equations
Now we have two equations:
1. \( 2A + 9D = 24 \) (Equation 1)
2. \( 2A + 19D = 44 \) (Equation 2)

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:
\[ D = 2 \]

### Conclusion
The common difference \( D \) of the arithmetic progression is:
\[ \boxed{2} \]