The interior angles of a polygon are in arithmetic progression. The smallest angle is 120°, and the

Q: Why do we get two answers for $n$? Does the math lie to us?

The math is perfectly honest; it is the geometry that imposes limits. The quadratic equation represents the sum of angles, but it does not inherently know about the 'convexity' rule. We must apply that constraint manually to filter out the physically impossible solution.

Q: What happens if I don't check the largest angle?

You would likely accept both $9$ and $16$ as valid answers, which would be incorrect. In JEE, always check if your algebraic solution satisfies the physical or geometric constraints of the problem. A polygon with an angle $\ge 180^\circ$ is concave, not convex.

Q: Is there a faster way to solve this?

Yes! You could use the sum of exterior angles, which is always $360^\circ$. Since the interior angles are in A.P., the exterior angles are also in A.P. This is a great alternative perspective to verify your result.

Question

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

Sigma Percentile · Accepted Answer

### The Geometry of Progressions

Imagine you are standing before a polygon, but not just any polygon. This one is special. Its interior angles are not random; they are dancing to the rhythm of an Arithmetic Progression.

The smallest angle starts at $120^\circ$, and each subsequent angle grows by a steady $5^\circ$. Our mission is to determine how many sides, $n$, this polygon possesses.

This problem is a classic JEE trap, designed to test whether you can bridge the gap between pure algebra and geometric reality.

### The Dual Nature of Summation

To solve this, we must look at the sum of the interior angles from two different perspectives.

First, we have the rigid, unyielding law of Euclidean geometry. For any $n$-sided polygon, the sum of interior angles is given by the formula:

$$S_n = (n - 2) 	imes 180^\circ$$

Second, we have the fluid nature of our Arithmetic Progression. The sum of the first $n$ terms of an A.P. is given by:

$$S_n = \frac{n}{2} [2a + (n - 1)d]$$

Here, $a = 120^\circ$ and $d = 5^\circ$. We now have two expressions for the same total sum. The bridge between them is simple equality.

### The Algebraic Bridge

Let us equate these two expressions:

$$\frac{n}{2} [2(120) + (n - 1)5] = (n - 2)180$$

Now, we must navigate the algebra with precision. Multiplying both sides by $2$ clears the fraction:

$$n [240 + 5n - 5] = 360(n - 2)$$

Simplifying the bracket, we get $n [235 + 5n] = 360n - 720$. Expanding this leads us to:

$$5n^2 + 235n = 360n - 720$$

Bringing everything to one side, we arrive at the quadratic equation:

$$5n^2 - 125n + 720 = 0$$

Dividing by $5$ simplifies our life significantly:

$$n^2 - 25n + 144 = 0$$

### The Quadratic Trap

Factorizing this quadratic is the next step. We need two numbers that multiply to $144$ and add to $-25$. Those numbers are $-9$ and $-16$.

Thus, we have:

$$(n - 9)(n - 16) = 0$$

This gives us two potential candidates: $n = 9$ or $n = 16$. Here is where the JEE examiner smiles, waiting to see if you will blindly accept both.

But we are smarter than that. We must apply the **Convexity Constraint**.

### The Geometric Reality Check

For a polygon to be convex, every single interior angle must be strictly less than $180^\circ$. Let us test our candidates.

The largest angle in our polygon is the $n$-th term of the A.P., given by $a_n = a + (n - 1)d$. For $n = 16$, the largest angle is:

$$a_{16} = 120 + 15(5) = 195^\circ$$

This is impossible! A $195^\circ$ angle makes the polygon concave, caving inward. We must reject $n = 16$.

Now, testing $n = 9$, the largest angle is:

$$a_9 = 120 + 8(5) = 160^\circ$$

Since $160^\circ < 180^\circ$, this is perfectly valid. The polygon has exactly **$9$ sides**.

Always remember: algebra provides the possibilities, but geometry dictates the truth.

Animated Solution for Mathematics - Sequence and Series: The interior angles of a polygon are in arithmetic progression. The smallest angle is $12 0^{\circ}$ , and the common difference is $5^{\circ}$ , Find the number of sides of the polygon.

Enter Numerical Value:

Visualized Solution

Visualizing the Polygon

The Geometric Sum Formula

The A.P. Sum Formula

Equating the Expressions

Simplifying the Equation

Expanding the Terms

Standard Quadratic Form

Factorizing the Quadratic

The Convexity Constraint

Testing $n = 16$

Testing $n = 9$

Final Conclusion

The Sigma Insight: Arithmetic Progression (A.P.)

The Geometry of Progressions

The Dual Nature of Summation

The Algebraic Bridge

The Quadratic Trap

The Geometric Reality Check

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