The interior angles of a convex polygon are in A.P. If the smallest angle is `100^(@)` and the common difference is `4^(@)`, then the number of sides is

Options

A. 5
B. 7
C. 36
D. 44

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understanding the Problem
We are given that the interior angles of a convex polygon are in an arithmetic progression (A.P.). The smallest angle is \(100^\circ\) and the common difference is \(4^\circ\). We need to find the number of sides \(n\) of the polygon.

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

### Step 3: Angles in A.P.
The angles in A.P. can be expressed as:
- First angle \(a = 100^\circ\)
- Second angle \(a + d = 100^\circ + 4^\circ = 104^\circ\)
- Third angle \(a + 2d = 100^\circ + 2 \times 4^\circ = 108^\circ\)
- ...
- \(n\)-th angle \(a + (n-1)d = 100^\circ + (n-1) \times 4^\circ\)

### Step 4: Sum of the Angles in A.P.
The sum of the angles can also be calculated using the formula for the sum of an A.P.:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Substituting the values of \(a\) and \(d\):
\[ S_n = \frac{n}{2} \times (2 \times 100 + (n-1) \times 4) \]
\[ S_n = \frac{n}{2} \times (200 + 4n - 4) = \frac{n}{2} \times (4n + 196) \]
\[ S_n = n(2n + 98) \]

### Step 5: Setting Up the Equation
We now set the two expressions for the sum of the interior angles equal to each other:
\[ n(2n + 98) = (n - 2) \times 180 \]

### Step 6: Simplifying the Equation
Expanding the right side:
\[ n(2n + 98) = 180n - 360 \]
Rearranging gives:
\[ 2n^2 + 98n - 180n + 360 = 0 \]
\[ 2n^2 - 82n + 360 = 0 \]
Dividing the entire equation by 2:
\[ n^2 - 41n + 180 = 0 \]

### Step 7: Solving the Quadratic Equation
Using the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 1\), \(b = -41\), and \(c = 180\):
\[ n = \frac{41 \pm \sqrt{(-41)^2 - 4 \times 1 \times 180}}{2 \times 1} \]
\[ n = \frac{41 \pm \sqrt{1681 - 720}}{2} \]
\[ n = \frac{41 \pm \sqrt{961}}{2} \]
\[ n = \frac{41 \pm 31}{2} \]

### Step 8: Finding the Values of \(n\)
Calculating the two possible values:
1. \(n = \frac{41 + 31}{2} = \frac{72}{2} = 36\)
2. \(n = \frac{41 - 31}{2} = \frac{10}{2} = 5\)

### Step 9: Validating the Values
Since the angles must be less than \(180^\circ\) for a convex polygon:
- For \(n = 36\): The largest angle is \(100 + 35 \times 4 = 240^\circ\) (not valid)
- For \(n = 5\): The largest angle is \(100 + 4 \times 4 = 116^\circ\) (valid)

Thus, the only valid solution is \(n = 5\).

### Final Answer
The number of sides of the polygon is \(5\).

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