Which term of the A.P. 4, 9, 14, 19 ...... is 109 ?

Options

A. (a) 12
B. (b) 19
C. (c) 22
D. (d) 27

Answer

To find which term of the arithmetic progression (A.P.) 4, 9, 14, 19, ... is equal to 109, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the first term (A)**:
The first term of the A.P. is given as:
\[ A = 4 \]

2. **Identify the common difference (D)**:
The common difference can be found by subtracting the first term from the second term:
\[ D = 9 - 4 = 5 \]

3. **Write the formula for the nth term (A_n)**:
The formula for the nth term of an A.P. is given by:
\[ A_n = A + (n - 1)D \]

4. **Set up the equation for the nth term equal to 109**:
We need to find the term where \( A_n = 109 \). Thus, we set up the equation:
\[ 109 = 4 + (n - 1) \cdot 5 \]

5. **Simplify the equation**:
Subtract 4 from both sides:
\[ 109 - 4 = (n - 1) \cdot 5 \]
\[ 105 = (n - 1) \cdot 5 \]

6. **Divide both sides by 5**:
\[ \frac{105}{5} = n - 1 \]
\[ 21 = n - 1 \]

7. **Solve for n**:
Add 1 to both sides:
\[ n = 21 + 1 \]
\[ n = 22 \]

### Conclusion:
The term 109 is the 22nd term of the A.P.