The difference any two consecutive interior angles of a polygon is 5∘.If the smallest angle is 120∘, find the number of the sides of the polygon.
We know that the sum of all angles of a polygon with n sides =1800(n−2)
We know that,
Sn=1800(n−2)........(1)
The angles of polygon will from an A.P.
Then common difference d=50
And first term =1200
let the side of polygon=n
Then,Sn=n2[2a+(n−1)d]........(2)
By equation (1) and (2) to,
n2[2a+(n−1)d]=1800(n−2)
⇒n2[2×1200+(n−1)50]=1800(n−2)
⇒n(2400+50n−50)=3600n−7200
⇒5n2+235n−360n+720=0
⇒5n2−125n+720=0
⇒5(n2−25n+144)=0
⇒n2−(16+9)n+144=0
⇒n2−16n−9n+144=0
⇒n(n−16)−9(n−16)=0
⇒(n−16)(n−9)=0
⇒n=9,16
Hence, this is the answer.