The interior angles of a polygon are in A.P. If the smallest angle is `120^@` and the common difference is `5^@` , then the number of sides in the polygon is :

Options

A. 8
B. 12
C. 9
D. 6

Answer

To solve the problem, we need to find the number of sides in a polygon whose interior angles are in an arithmetic progression (A.P.). Given that the smallest angle is \(120^\circ\) and the common difference is \(5^\circ\), we can follow these steps:

### Step 1: Understand the Sum of Interior Angles
The sum of the interior angles of a polygon with \(n\) sides is given by the formula:
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]

### Step 2: Define the Angles in A.P.
Since the angles are in A.P., we can express them as follows:
- First angle (smallest angle) = \(a = 120^\circ\)
- Second angle = \(a + d = 120 + 5 = 125^\circ\)
- Third angle = \(a + 2d = 120 + 2 \times 5 = 130^\circ\)
- And so on, until the \(n\)-th angle = \(a + (n-1)d = 120 + (n-1) \times 5\)

### Step 3: Write the Sum of the Angles in A.P.
The sum of the first \(n\) angles in A.P. can be calculated using the formula for the sum of an A.P.:
\[ \text{Sum} = \frac{n}{2} \times (2a + (n - 1)d) \]
Substituting \(a = 120^\circ\) and \(d = 5^\circ\):
\[ \text{Sum} = \frac{n}{2} \times (2 \times 120 + (n - 1) \times 5) \]
\[ = \frac{n}{2} \times (240 + 5n - 5) \]
\[ = \frac{n}{2} \times (5n + 235) \]

### Step 4: Set the Two Expressions for the Sum Equal
Now we set the sum of the interior angles equal to the sum calculated from the A.P.:
\[ (n - 2) \times 180 = \frac{n}{2} \times (5n + 235) \]

### 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: Solve the Quadratic Equation
Dividing the entire equation by 5:
\[ n^2 - 25n + 144 = 0 \]
Now we can use the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -25\), and \(c = 144\):
\[ n = \frac{25 \pm \sqrt{(-25)^2 - 4 \times 1 \times 144}}{2 \times 1} \]
\[ = \frac{25 \pm \sqrt{625 - 576}}{2} \]
\[ = \frac{25 \pm \sqrt{49}}{2} \]
\[ = \frac{25 \pm 7}{2} \]
Calculating the two possible values:
1. \(n = \frac{32}{2} = 16\)
2. \(n = \frac{18}{2} = 9\)

### Step 7: Determine Validity of \(n\)
We need to check which value of \(n\) is valid:
- For \(n = 16\): The angles would extend beyond \(180^\circ\) (as the last angle would be \(120 + 15 \times 5 = 195^\circ\)), which is not possible.
- For \(n = 9\): The angles would be \(120^\circ, 125^\circ, \ldots, 160^\circ\), all valid.

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