The interior angle of a convex polygon are in AP. The smallest angle is `(2 pi)/(3)` and the common difference is `5^(@)`. Then, the number of sides of the polygon are

Options

A. 16
B. 9
C. 12
D. 18

Answer

To find the number of sides of a convex polygon whose interior angles are in arithmetic progression (AP), we follow these steps:

### Step 1: Define the Variables
Let:
- \( n \) = number of sides of the polygon
- \( a \) = smallest angle = \( \frac{2\pi}{3} \) radians = 120 degrees
- \( d \) = common difference = 5 degrees

### Step 2: Sum of Interior Angles of a Polygon
The sum of the interior angles of a polygon with \( n \) sides is given by the formula:
\[ \text{Sum of angles} = (n - 2) \times 180 \text{ degrees} \]

### Step 3: Sum of Angles in AP
The sum of the angles in an arithmetic progression can also be expressed as:
\[ \text{Sum of angles} = \frac{n}{2} \times (2a + (n - 1)d) \]
Substituting \( a = 120 \) degrees and \( d = 5 \) degrees, we have:
\[ \text{Sum of angles} = \frac{n}{2} \times (2 \times 120 + (n - 1) \times 5) \]
\[ = \frac{n}{2} \times (240 + 5n - 5) \]
\[ = \frac{n}{2} \times (5n + 235) \]

### Step 4: Set the Two Expressions for Sum Equal
Set the two expressions for the sum of angles equal to each other:
\[ (n - 2) \times 180 = \frac{n}{2} \times (5n + 235) \]

### Step 5: Simplify the Equation
Expanding both sides:
\[ 180n - 360 = \frac{n(5n + 235)}{2} \]
Multiplying through by 2 to eliminate the fraction:
\[ 360n - 720 = n(5n + 235) \]
\[ 360n - 720 = 5n^2 + 235n \]
Rearranging gives:
\[ 5n^2 - 125n + 720 = 0 \]

### Step 6: Divide the Equation by 5
To simplify, divide the entire equation by 5:
\[ n^2 - 25n + 144 = 0 \]

### Step 7: Factor the Quadratic Equation
Now we factor the quadratic:
\[ (n - 16)(n - 9) = 0 \]
Thus, the solutions for \( n \) are:
\[ n = 16 \quad \text{or} \quad n = 9 \]

### Step 8: Check for Validity
Since the polygon is convex, the largest angle must be less than 180 degrees.

For \( n = 16 \):
- The largest angle would be:
\[ a + (n - 1)d = 120 + 15 \times 5 = 120 + 75 = 195 \text{ degrees (not valid)} \]

For \( n = 9 \):
- The largest angle would be:
\[ a + (n - 1)d = 120 + 8 \times 5 = 120 + 40 = 160 \text{ degrees (valid)} \]

### Conclusion
Thus, the number of sides of the polygon is:
\[ \boxed{9} \]