If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is

Options

A. `87`
B. `88`
C. `89`
D. `90`

Answer

To find the 18th term of the arithmetic progression (A.P.) given that the 7th term is 34 and the 13th term is 64, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for the nth term of an A.P.**:
The nth term of an A.P. can be expressed as:
\[ A_n = A + (n-1)d \]
where \( A \) is the first term and \( d \) is the common difference.

2. **Set up the equations for the given terms**:
- For the 7th term:
\[ A_7 = A + 6d = 34 \quad \text{(1)} \]
- For the 13th term:
\[ A_{13} = A + 12d = 64 \quad \text{(2)} \]

3. **Subtract equation (1) from equation (2)**:
\[ (A + 12d) - (A + 6d) = 64 - 34 \]
Simplifying this gives:
\[ 12d - 6d = 30 \implies 6d = 30 \]
Therefore, we find:
\[ d = 5 \quad \text{(3)} \]

4. **Substitute \( d \) back into equation (1) to find \( A \)**:
\[ A + 6(5) = 34 \]
Simplifying this gives:
\[ A + 30 = 34 \implies A = 4 \quad \text{(4)} \]

5. **Now, find the 18th term using the values of \( A \) and \( d \)**:
\[ A_{18} = A + 17d \]
Substituting the values from (3) and (4):
\[ A_{18} = 4 + 17(5) = 4 + 85 = 89 \]

### Final Answer:
The 18th term of the A.P. is \( \boxed{89} \).