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: Write the formulas for the 2nd and 5th terms of the AP.
The nth term of an AP can be expressed as:
\[ A_n = A + (n - 1)D \]
where \( A \) is the first term, \( D \) is the common difference, and \( n \) is the term number.

For the 2nd term:
\[ A_2 = A + (2 - 1)D = A + D \]
Given \( A_2 = 13 \):
\[ A + D = 13 \quad \text{(Equation 1)} \]

For the 5th term:
\[ A_5 = A + (5 - 1)D = A + 4D \]
Given \( A_5 = 25 \):
\[ A + 4D = 25 \quad \text{(Equation 2)} \]

### Step 2: Solve the equations simultaneously.
From Equation 1:
\[ A + D = 13 \]
We can express \( A \) in terms of \( D \):
\[ A = 13 - D \quad \text{(Equation 3)} \]

Now, substitute Equation 3 into Equation 2:
\[ (13 - D) + 4D = 25 \]
This simplifies to:
\[ 13 - D + 4D = 25 \]
\[ 13 + 3D = 25 \]

### Step 3: Isolate \( D \).
Subtract 13 from both sides:
\[ 3D = 25 - 13 \]
\[ 3D = 12 \]
Now divide by 3:
\[ D = \frac{12}{3} = 4 \]

### Step 4: Find \( A \) using the value of \( D \).
Substituting \( D = 4 \) back into Equation 3:
\[ A = 13 - D = 13 - 4 = 9 \]

### Step 5: Calculate the 7th term \( A_7 \).
Using the formula for the nth term:
\[ A_7 = A + (7 - 1)D \]
Substituting the values of \( A \) and \( D \):
\[ A_7 = 9 + (6)(4) \]
\[ A_7 = 9 + 24 = 33 \]

Thus, the 7th term of the AP is **33**.