The interior angles of a convex polygon are in arithmetic progression. The smallest angle is 120 degrees and the common difference is 5 degrees. Find the number of sides of the polygon.
+129852 Let the first term be 120
Let the last term = 120 + 5(n - 1)
The sum of the interior angles = (number of sides - 2) 180
Sum of the interior angles = [ 1st term + last term ] (number of terms/2 ]
First term =120
Last term = 120 + 5(n - 1)
So
[120 + 120 + 5(n-1) ] (n /2) = (n - 2) 180
[ 240 + 5n - 5 ] n = (n - 2) 360
[ 235 + 5n ] n = (n - 2)360
235n + 5n^2 = 360n - 720 rearrange as
5n^2 - 125n + 720 = 0 divide through by 5
n^2 - 25n + 144 = 0 factor as
(n - 16) ( n - 9) = 0
We have two
Set each factor to 0 and solve for n
n = 16 or n = 9
We have two possibilities
n = 9 or n = 16
+285 I got this answer from https://www.topperlearning.com/answer/the-interior-angles-of-a-polygon-are-in-apif-the-smallest-angle-is-120-degree-and-the-common-difference-is-5-degree-then-the-no-of-sides-in-the-polygo/tk5eo455 so check it out
Smallest angle=120degrees
Common difference=5
A P is 120, 125, 130, ..
The sum of interior angles of a polygon= (n-2)180
Hence Sum of n terms of an A P = (n-2)180
n/2 {2.120+(n-1)5} = 180(n-2)
5n2 -125n +720 = 0
n2 -25n +144=0
n=9 or 16
Hence number of sides can be 9 or 16
-hihihi
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