The interior angles of a polygon are in arithmetic progression. The smallest angle is `120^(@)` and the common difference is 5. The number of sides of the polygon is

Options

A. 7
B. 9
C. 11
D. 16

Answer

To solve the problem, we need to find the number of sides of a polygon where the interior angles are in arithmetic progression, with the smallest angle being \(120^\circ\) and a common difference of \(5^\circ\).

### Step-by-step Solution:

1. **Identify the formula for the sum of interior angles**:
The sum of the interior angles of a polygon with \(n\) sides is given by the formula:
\[ S = (n - 2) \times 180^\circ \]

2. **Define the angles in terms of \(n\)**:
Since the angles are in arithmetic progression, we can express the angles as:
- First angle (smallest): \(a = 120^\circ\)
- Common difference: \(d = 5^\circ\)
- The angles can be expressed as:
\[ 120^\circ, 125^\circ, 130^\circ, \ldots, (120 + (n-1) \times 5) \]

3. **Sum of the interior angles in terms of \(n\)**:
The sum of the \(n\) angles can be calculated using the formula for the sum of an arithmetic series:
\[ S = \frac{n}{2} \times (2a + (n - 1)d) \]
Substituting \(a = 120^\circ\) and \(d = 5^\circ\):
\[ S = \frac{n}{2} \times (2 \times 120 + (n - 1) \times 5) \]
Simplifying this gives:
\[ S = \frac{n}{2} \times (240 + 5n - 5) = \frac{n}{2} \times (5n + 235) \]

4. **Set the two expressions for the sum equal**:
We set the sum of the angles from the arithmetic progression equal to the sum of the interior angles formula:
\[ \frac{n}{2} \times (5n + 235) = (n - 2) \times 180 \]

5. **Clear the fraction by multiplying through by 2**:
\[ n(5n + 235) = 2(n - 2) \times 180 \]
This simplifies to:
\[ 5n^2 + 235n = 360n - 720 \]

6. **Rearranging the equation**:
Bringing all terms to one side gives:
\[ 5n^2 + 235n - 360n + 720 = 0 \]
Which simplifies to:
\[ 5n^2 - 125n + 720 = 0 \]

7. **Divide the entire equation by 5**:
\[ n^2 - 25n + 144 = 0 \]

8. **Use the quadratic formula to solve for \(n\)**:
The quadratic formula is:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -25\), and \(c = 144\):
\[ n = \frac{25 \pm \sqrt{(-25)^2 - 4 \times 1 \times 144}}{2 \times 1} \]
\[ n = \frac{25 \pm \sqrt{625 - 576}}{2} \]
\[ n = \frac{25 \pm \sqrt{49}}{2} \]
\[ n = \frac{25 \pm 7}{2} \]

9. **Calculate the possible values of \(n\)**:
\[ n = \frac{32}{2} = 16 \quad \text{and} \quad n = \frac{18}{2} = 9 \]

10. **Check the validity of the solutions**:
- For \(n = 16\):
The largest angle would be \(120 + (16-1) \times 5 = 120 + 75 = 195^\circ\), which is not valid since angles must be less than \(180^\circ\).
- For \(n = 9\):
The largest angle would be \(120 + (9-1) \times 5 = 120 + 40 = 160^\circ\), which is valid.

Thus, the number of sides of the polygon is \(n = 9\).

### Final Answer:
The number of sides of the polygon is \(9\).