If the 3rd and 7th terms of an AP are 17 and 27 respectively find the fifth term of the AP:

Options

A. 9
B. 22
C. 14
D. none of these

Answer

To solve the problem, we need to find the fifth term of an arithmetic progression (AP) given that the third term is 17 and the seventh term is 27.

### Step-by-Step Solution:

1. **Understanding the nth term of an AP**:
The nth term of an arithmetic progression 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 problem, we know:
- The 3rd term \( T_3 = 17 \):
\[ T_3 = a + (3-1)d = a + 2d = 17 \quad \text{(Equation 1)} \]
- The 7th term \( T_7 = 27 \):
\[ T_7 = a + (7-1)d = a + 6d = 27 \quad \text{(Equation 2)} \]

3. **Subtracting the equations**:
To eliminate \( a \), we can subtract Equation 1 from Equation 2:
\[ (a + 6d) - (a + 2d) = 27 - 17 \]
Simplifying this gives:
\[ 6d - 2d = 10 \implies 4d = 10 \implies d = \frac{10}{4} = \frac{5}{2} \]

4. **Finding the first term \( a \)**:
Now that we have \( d \), we can substitute it back into Equation 1 to find \( a \):
\[ a + 2d = 17 \]
Substituting \( d = \frac{5}{2} \):
\[ a + 2 \left(\frac{5}{2}\right) = 17 \]
This simplifies to:
\[ a + 5 = 17 \implies a = 17 - 5 = 12 \]

5. **Finding the fifth term \( T_5 \)**:
Now we can find the fifth term using the formula for the nth term:
\[ T_5 = a + (5-1)d = a + 4d \]
Substituting the values of \( a \) and \( d \):
\[ T_5 = 12 + 4 \left(\frac{5}{2}\right) \]
This simplifies to:
\[ T_5 = 12 + 10 = 22 \]

### Final Answer:
The fifth term of the AP is \( \boxed{22} \).