If the 2nd term of an AP is 13 and 5th term is 25, what is its 7th term ?

Options

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

Explanation

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

### Step 1: Define the terms of the AP
Let the first term of the AP be \( A_1 \) and the common difference be \( D \). The terms of the AP can be expressed as:
- \( A_2 = A_1 + D \)
- \( A_5 = A_1 + 4D \)

### Step 2: Set up the equations
From the information given:
- \( A_2 = 13 \) implies \( A_1 + D = 13 \) (Equation 1)
- \( A_5 = 25 \) implies \( A_1 + 4D = 25 \) (Equation 2)

### Step 3: Solve the equations
Now, we will solve these two equations simultaneously.

1. From Equation 1:
\[ A_1 + D = 13 \implies D = 13 - A_1 \]

2. Substitute \( D \) in Equation 2:
\[ A_1 + 4(13 - A_1) = 25 \]
Simplifying this gives:
\[ A_1 + 52 - 4A_1 = 25 \]
\[ -3A_1 + 52 = 25 \]
\[ -3A_1 = 25 - 52 \]
\[ -3A_1 = -27 \implies A_1 = 9 \]

3. Now substitute \( A_1 \) back to find \( D \):
\[ D = 13 - A_1 = 13 - 9 = 4 \]

### Step 4: Find the 7th term
Now that we have \( A_1 = 9 \) and \( D = 4 \), we can find the 7th term \( A_7 \) using the formula for the nth term of an AP:
\[ A_n = A_1 + (n - 1)D \]
For \( n = 7 \):
\[ A_7 = A_1 + (7 - 1)D = 9 + 6 \times 4 \]
Calculating this gives:
\[ A_7 = 9 + 24 = 33 \]

### Final Answer
The 7th term of the AP is \( \boxed{33} \).