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# Polygon

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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.

Apr 12, 2022

#1
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Sum  of this arithmetic series  =    n   ( a1  + an)   / 2

n = number of sides

a1= 120  = first term

an  = 120 + 5 (n -1) = 115 + 5n   =  last term

So  we  have    n [ 235 + 5n ] / 2      (1)

And the sum of the interior angles of an n-sided polygon =    (n - 2) (180)     (2)

Set (1)  = (2)

n [ 235 + 5n ] /2  = (n -2) (180)    simplify

235n + 5n^2  = 2 ( n -2)(180)

235n + 5n^2  = (n -2) (360)

235n + 5n^2  = 360n - 720

5n^2 - 125n + 720  =  0            divide through by 5

n^2 - 25n + 144   = 0          factor

(n - 16) ( n - 9)   = 0

Setting  each factor  to  0 and solve for n and we get two answers

n = 9     or  n =  16   Apr 12, 2022
edited by CPhill  Apr 12, 2022
#2
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N = 16 takes the interior angle past 180 degrees doesn't it ?

Apr 12, 2022
#3
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Yep, Guest,  you  are correct.....thanks for pointing that out  !!!!

I forgot that the polygon is  convex.....n =16 would produce a concave polygon !!!   