If the third term of an A.P. is 12 and 10th term is 26, then its 20th term is :

Options

A. 46
B. 52
C. 50
D. `44`

Answer

To solve the problem, we need to find the 20th term of an Arithmetic Progression (A.P.) given that the third term is 12 and the tenth term is 26.

### Step-by-Step Solution:

1. **Understanding the nth Term Formula**:
The nth term of an A.P. can be expressed as:
\[ T_n = A + (n - 1)D \]
where \( T_n \) is the nth term, \( A \) is the first term, \( D \) is the common difference, and \( n \) is the term number.

2. **Setting Up the Equations**:
- For the 3rd term (\( n = 3 \)):
\[ T_3 = A + (3 - 1)D = A + 2D = 12 \quad \text{(Equation 1)} \]
- For the 10th term (\( n = 10 \)):
\[ T_{10} = A + (10 - 1)D = A + 9D = 26 \quad \text{(Equation 2)} \]

3. **Solving the Equations**:
Now we have two equations:
- Equation 1: \( A + 2D = 12 \)
- Equation 2: \( A + 9D = 26 \)

We can subtract Equation 1 from Equation 2 to eliminate \( A \):
\[ (A + 9D) - (A + 2D) = 26 - 12 \]
Simplifying this gives:
\[ 7D = 14 \]
Therefore, we find:
\[ D = 2 \]

4. **Finding the First Term \( A \)**:
Now we can substitute \( D \) back into Equation 1 to find \( A \):
\[ A + 2(2) = 12 \]
This simplifies to:
\[ A + 4 = 12 \]
Thus:
\[ A = 12 - 4 = 8 \]

5. **Finding the 20th Term**:
Now that we have \( A \) and \( D \), we can find the 20th term:
\[ T_{20} = A + (20 - 1)D = A + 19D \]
Substituting the values of \( A \) and \( D \):
\[ T_{20} = 8 + 19(2) = 8 + 38 = 46 \]

### Final Answer:
The 20th term of the A.P. is **46**.

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