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

Explanation

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

### Step 1: Define the variables
Let the number of sides of the polygon be \( n \). The smallest angle is given as \( 120^\circ \) and the common difference of the angles in the arithmetic progression is \( 5^\circ \).

### Step 2: Write 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:
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]

### Step 3: Write the expression for the sum of the angles in AP
Since the angles are in arithmetic progression, the angles can be expressed as:
- First angle: \( 120^\circ \)
- Second angle: \( 120 + 5 = 125^\circ \)
- Third angle: \( 120 + 10 = 130^\circ \)
- ...
- \( n \)-th angle: \( 120 + (n - 1) \times 5 \)

The sum of the angles in arithmetic progression can be calculated using the formula:
\[ \text{Sum} = \frac{n}{2} \times (\text{First term} + \text{Last term}) \]
Where the last term is \( 120 + (n - 1) \times 5 \).

Thus, the sum becomes:
\[ \text{Sum} = \frac{n}{2} \times \left(120 + \left(120 + (n - 1) \times 5\right)\right) \]
This simplifies to:
\[ \text{Sum} = \frac{n}{2} \times \left(240 + (n - 1) \times 5\right) \]
\[ = \frac{n}{2} \times (240 + 5n - 5) \]
\[ = \frac{n}{2} \times (5n + 235) \]

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

### Step 5: Simplify the equation
Multiply both sides by 2 to eliminate the fraction:
\[ n(5n + 235) = 2(n - 2) \times 180 \]
\[ n(5n + 235) = 360n - 720 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 5n^2 + 235n - 360n + 720 = 0 \]
\[ 5n^2 - 125n + 720 = 0 \]

### Step 7: Divide the equation by 5
\[ n^2 - 25n + 144 = 0 \]

### Step 8: Factor the quadratic equation
Factoring the quadratic:
\[ (n - 16)(n - 9) = 0 \]

### Step 9: Solve for \( n \)
Setting each factor to zero gives:
\[ n - 16 = 0 \quad \Rightarrow \quad n = 16 \]
\[ n - 9 = 0 \quad \Rightarrow \quad n = 9 \]

### Step 10: Determine the valid solution
If \( n = 16 \):
- The largest angle would be \( 120 + 15 \times 5 = 195^\circ \), which is not possible for a polygon.

If \( n = 9 \):
- The angles would be \( 120^\circ, 125^\circ, 130^\circ, \ldots, 160^\circ \), all of which are valid.

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