Problem 258 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

The interior angles of a polygon are in A.P., the least angle is \(120^{\circ}\) and the common difference is \(5^{\circ} .\) Find the number of sides.

Step-by-Step Solution

Answer

The number of sides of the polygon is 12.

Step 1

Define the general term of the A.P. In the A.P., the nth term formula is: \( a_n = a + (n-1) * d \), where 'a' is the first term, 'n' is the term number, and 'd' is the common difference. Substituting the given values, we get \( a_n = 120 + (n-1) * 5 \), where n is the term number.

Step 2

Find the sum of angles in the polygon. The sum of interior angles of a polygon is given by \( (n-2) * 180 \), where n is the side count of the polygon.

Step 3

Equate the sum of A.P. to the sum of polygon angles. The sum of first n terms in an AP is given by \( S = n/2 * [2a + (n-1) * d] \), substituting the given values, we get \( S = n/2 * [2*120 + (n-1) * 5] \). Equating this to the formula \( (n-2) * 180 \), we are ready to solve for n.

Step 4

Upon solving the equation, n turns out to be 12.

Key Concepts

Arithmetic Progression

Understanding arithmetic progression (AP) is crucial when studying patterns in mathematics, especially in sequences. An AP is a sequence of numbers in which the difference between the consecutive terms is constant. This difference is known as the common difference, denoted by 'd'. The AP is defined by its first term 'a' and the common difference 'd'.

For instance, in the problem where the interior angles of a polygon are in AP with the least angle being 120 degrees and the common difference being 5 degrees, we can represent the angles as 120°, 125°, 130°, and so on. To find the nth term of an AP, the formula used is:
\[ a_n = a + (n-1) * d \]
In this case, by substituting into the formula, we establish the general term of the sequence. The understanding of AP is not only vital for this problem but also forms the foundation for more complex mathematical sequences and series.

Polygon Angle Sum

The polygon angle sum is the total measure of all interior angles inside a polygon. The formula for calculating this sum is pivotal when dealing with polygonal shapes. For an n-sided polygon, the sum of the interior angles can be found by:
\[ (n-2) * 180^\circ \]
This formula is derived from the fact that any polygon can be divided into triangles, and since each triangle has an angle sum of 180 degrees, the polygon's angle sum is essentially the sum of the interior angles of the triangles that make it up.

In our problem, we use the polygon angle sum formula to establish a relationship with the AP sum. The polygon angle sum provides an important clue to unlocking the number of sides of the polygon when paired with the information about the interior angles being in arithmetic progression.

Interior Angle Formula

The interior angle formula is a direct application of the polygon angle sum and helps us to find the number of sides of a polygon based on its angles. When we know that the interior angles are in an arithmetic progression, we can also use the sum formula for AP:
\[ S = \frac{n}{2} * [2a + (n-1) * d] \]
Here 'S' represents the sum of the first 'n' terms of an AP, 'a' is the first term, and 'd' is the common difference. By equating this sum to the polygon angle sum formula, we can solve for 'n', the number of sides.

In our example, after setting up the equation based on the given least angle of 120 degrees and common difference of 5 degrees, we bridge the gap between AP and polygon properties. This is where we integrate both AP and polygon angle sum to unravel the mystery - the number of sides, which, upon calculation, reveals itself to be 12 for the given polygon.