If the `2^(nd)` term of an A.P. is 13 and the `5^(th)` term is 25, what is its `7 ^(th)` term ?

Options

A. 30
B. 33
C. 37
D. 38

Answer

To find the 7th term of the arithmetic progression (A.P.) given that the 2nd term is 13 and the 5th term is 25, we can follow these steps:

### Step-by-Step Solution:

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

2. **Setting up the equations**:
From the information given:
- For the 2nd term \( T_2 \):
\[ T_2 = A + (2-1)D = A + D = 13 \quad \text{(1)} \]
- For the 5th term \( T_5 \):
\[ T_5 = A + (5-1)D = A + 4D = 25 \quad \text{(2)} \]

3. **Subtracting the equations**:
We can eliminate \( A \) by subtracting equation (1) from equation (2):
\[ (A + 4D) - (A + D) = 25 - 13 \]
Simplifying this gives:
\[ 4D - D = 12 \implies 3D = 12 \implies D = 4 \]

4. **Finding the first term \( A \)**:
Substitute \( D = 4 \) back into equation (1):
\[ A + 4 = 13 \implies A = 13 - 4 = 9 \]

5. **Finding the 7th term \( T_7 \)**:
Now we can find the 7th term using the formula:
\[ T_7 = A + (7-1)D = A + 6D \]
Substituting the values of \( A \) and \( D \):
\[ T_7 = 9 + 6 \times 4 = 9 + 24 = 33 \]

### Final Answer:
The 7th term of the A.P. is **33**.