Question

The interior angles of a polygon with \( n \) sides, are in an A.P. with common difference 6°. If the largest interior angle of the polygon is 219°, then \( n \) is equal to:

• A. 20
• B. 18
• C. 25
• D. 15

Hint:

When dealing with arithmetic progressions in geometry, use the standard formulas for sum and difference of angles to set up and solve equations.

Answer 1

The Correct Option is A

Step 1: Define the Given Information 

We are given that the interior angles of a polygon are in arithmetic progression (A.P.) with a common difference of 6° and the largest angle is 219°. 
The sum of the interior angles of an \( n \)-sided polygon is given by: \[ \frac{n}{2} \left( 2a + (n-1) \times 6 \right) = (n-2) \times 180 \] where \( a \) is the first angle. Simplifying: \[ an + 3n^2 - 3n = (n-2) \times 180 \]

Step 2: Apply the Condition for the Largest Angle

We are also given that the largest interior angle is 219°, which gives the equation: \[ a + (n-1) \times 6 = 219 \] Simplifying this equation: \[ a = 225 - 6n \]

Step 3: Substitute into the Sum Equation

Now, substitute the value of \( a = 225 - 6n \) into the sum equation: \[ (225 - 6n) + 3n^2 - 3n = (n-2) \times 180 \] Simplifying and solving this quadratic equation, we get: \[ n = 20 \]

Final Answer: \( n = 20 \)

Answer 2

Step 1: Understanding the problem.
We are given a polygon with \( n \) sides whose interior angles are in an arithmetic progression (A.P.) with a common difference of \( 6^\circ \). The largest interior angle is \( 219^\circ \). We need to find the number of sides \( n \).

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

Step 3: Express angles in A.P. form.
Let the smallest angle be \( a \).
Then, the \( n \) angles are:
\[ a, \, (a + 6), \, (a + 12), \, (a + 18), \, \dots, \, [a + (n - 1) \times 6]. \]
The largest angle is \( a + (n - 1) \times 6 = 219^\circ. \)

Step 4: Write the equation for the sum of angles.
The sum of all interior angles = sum of n terms of an A.P.:
\[ \frac{n}{2} [2a + (n - 1) \times 6] = (n - 2) \times 180. \]

Step 5: Substitute \( a + (n - 1) \times 6 = 219 \).
\[ a = 219 - 6(n - 1) = 225 - 6n. \] Substitute this value into the sum equation:
\[ \frac{n}{2} [2(225 - 6n) + (n - 1) \times 6] = (n - 2) \times 180. \]

Step 6: Simplify the equation.
\[ \frac{n}{2} [450 - 12n + 6n - 6] = (n - 2) \times 180. \] \[ \frac{n}{2} (444 - 6n) = 180n - 360. \] Multiply both sides by 2:
\[ n(444 - 6n) = 360n - 720. \] Simplify:
\[ 444n - 6n^2 = 360n - 720. \] \[ 84n = 6n^2 - 720. \] \[ 6n^2 - 84n - 720 = 0. \] Divide by 6:
\[ n^2 - 14n - 120 = 0. \]

Step 7: Solve the quadratic equation.
\[ n = \frac{14 \pm \sqrt{14^2 + 480}}{2} = \frac{14 \pm \sqrt{676}}{2} = \frac{14 \pm 26}{2}. \] So, \( n = 20 \) or \( n = -6 \). Since \( n \) must be positive, \( n = 20. \)

Final Answer:
\[ \boxed{n = 20} \]