If the first and the last term is an AP are 17 and 350 respectively and the common difference is 9 , then the number of terms is .

Options

A. 38
B. 35
C. 30
D. 40

Answer

To solve the problem, we need to find the number of terms in an arithmetic progression (AP) where the first term (A) is 17, the last term (an) is 350, and the common difference (D) is 9.

### Step-by-Step Solution:

1. **Understand the formula for the nth term of an AP**:
The nth term of an arithmetic progression can be expressed as:
\[ a_n = A + (n - 1)D \]
where:
- \( a_n \) is the nth term (last term in this case),
- \( A \) is the first term,
- \( D \) is the common difference,
- \( n \) is the number of terms.

2. **Substitute the known values**:
Here, we know:
- \( a_n = 350 \)
- \( A = 17 \)
- \( D = 9 \)

Plugging these values into the formula gives:
\[ 350 = 17 + (n - 1) \cdot 9 \]

3. **Rearrange the equation**:
To isolate \( n \), first subtract 17 from both sides:
\[ 350 - 17 = (n - 1) \cdot 9 \]
This simplifies to:
\[ 333 = (n - 1) \cdot 9 \]

4. **Divide both sides by 9**:
Now, divide both sides by 9 to solve for \( n - 1 \):
\[ n - 1 = \frac{333}{9} \]

5. **Calculate \( \frac{333}{9} \)**:
Performing the division:
\[ \frac{333}{9} = 37 \]
Thus, we have:
\[ n - 1 = 37 \]

6. **Solve for \( n \)**:
Finally, add 1 to both sides to find \( n \):
\[ n = 37 + 1 = 38 \]

### Conclusion:
The number of terms in the arithmetic progression is \( n = 38 \).