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 an arithmetic progression (AP) where 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:
\[ a_n = a + (n - 1) \cdot d \]
Where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

For the 2nd term:
\[ a_2 = a + (2 - 1) \cdot d = a + d \]
Given that \( a_2 = 13 \), we can write:
\[ a + d = 13 \quad \text{(Equation 1)} \]

For the 5th term:
\[ a_5 = a + (5 - 1) \cdot d = a + 4d \]
Given that \( a_5 = 25 \), we can write:
\[ a + 4d = 25 \quad \text{(Equation 2)} \]

### Step 2: Subtract Equation 1 from Equation 2
Now, we will subtract Equation 1 from Equation 2 to eliminate \( a \):
\[ (a + 4d) - (a + d) = 25 - 13 \]
This simplifies to:
\[ 4d - d = 12 \]
\[ 3d = 12 \]

### Step 3: Solve for \( d \)
Now, divide both sides by 3:
\[ d = \frac{12}{3} = 4 \]

### Step 4: Substitute \( d \) back into Equation 1 to find \( a \)
Now that we have \( d \), we can substitute it back into Equation 1:
\[ a + d = 13 \]
\[ a + 4 = 13 \]
Subtract 4 from both sides:
\[ a = 13 - 4 = 9 \]

### Step 5: Find the 7th term
Now we can find the 7th term using the formula for the nth term:
\[ a_7 = a + (7 - 1) \cdot d = a + 6d \]
Substituting the values of \( a \) and \( d \):
\[ a_7 = 9 + 6 \cdot 4 \]
\[ a_7 = 9 + 24 = 33 \]

### Conclusion
The 7th term of the AP is \( 33 \).

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