The Interior Angles of a Polygon are in Arithmetic Progression GMAT Problem Solving

Question: The interior angles of a polygon are in arithmetic progression. If the smallest angle is 120° and the common difference is 5°, what is the number of sides of polygon ?

1. 4
2. 6
3. 9
4. 12
5. 16

Correct Answer:C 

Solution and Explanation
Approach Solution 1:
Given: The Smallest angle is a=120 and the common difference between angles is 5.

Candidates need to find out how many sides of a polygon are present

Common Difference = 5
As per the arithmetic equation,
Sum of interior angles of a polygon = (n-2)x180

Arithmetic Progression, Sum = (n/2) x (2a+(n-1)d)
(n/2) x { (2x120) + (n-1)x5 } = (n-2)x180
(n) x (240 + 5n - 5) = (n-2)x360
240n + 5n^2 - 5n = 360n - 720
5n^2 + 235n - 360n + 720 = 0
5n^2 - 125n + 720 = 0
n^2 - 25n + 144 = 0
n^2 - 9n - 16n + 144 = 0
(n-9)(n-16) = 0
n = 9 ; 16
There are two options which are 9 and 16
n = 16 is not possible , as it will be greater than 180.

Hence, the correct answer will be C.

Approach Solution 2:
Let's solve it by using the options.

Let’s start by option C i.e 9.(Make it a habit to begin with C in GMAT as the options are arranged in ascending or descending order. This may not help everywhere (like in this case). However, building up this practice habit will provide dividends in the long run, especially in questions that can be solved using the technique of back-solving.
If the number of sides is 9, then the sum of all the angles is (n-2) * 180 = 7 * 180 = 1260 degrees.

With the smallest angle being 120,and the common difference being 5,the angles would be 120,125,130,135, 140, 145, 150, 155, 160 ....

Thus sum of such angles = 9/2(2*120 + 40) using Sum = (n/2) * (2a+(n-1)d) in an AP where a= first term; n= total number of terms; d=common difference

= 280 * 9 /2
=1260 which is the same as what we obtained earlier.

Hence, the correct answer is C..

Approach Solution 3:
Smallest angle, a = 120
C.D = 5
sum of interior angles of a polygon = (n-2)x180

AP, Sum = (n/2) x (2a+(n-1)d)
(n/2) x { (2x120) + (n-1)x5 } = (n-2)x180
(n) x (240 + 5n - 5) = (n-2)x360
240n + 5n^2 - 5n = 360n - 720
5n^2 + 235n - 360n + 720 = 0
5n^2 - 125n + 720 = 0
n^2 - 25n + 144 = 0
n^2 - 9n - 16n + 144 = 0
(n-9)(n-16) = 0
n = 9 ; 16
n = 16 is not possible , as it will be greater than 180.

“The interior angles of a polygon are in arithmetic progression.”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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