Answer :

Given: the sum of interior angles of a triangle is 180°


To prove: Sum of interior angles of polygons with 3, 4, 5,…… forms an A.P.


To find: Sum of interior angles for a 21 sided polygon i.e. a21


We know,


The sum of interior angles of a polygon with n sides,


an = 180° × (n – 2)


Sum of interior angles of a polygon of 3 sides,


a3= 180° × (3 – 2) = 180° × 1 = 180°


Sum of interior angles of a polygon of 4 sides,


a4= 180° × (4 – 2) = 180° × 2 = 360°


Sum of interior angles of a polygon of 5 sides,


a5= 180° × (5 – 2) = 180° × 3 = 540°


A.P is known for Arithmetic Progression whose common difference, d = an – an-1 where n > 0


Here,


d = a4 – a3 = 360° - 180° = 180°


d = a5 – a4 = 540° - 360° = 180°


Common difference is same in both the cases


This shows that sum of interior angles of polygons with 3, 4, 5,…… forms an A.P.


Hence Proved


Sum of interior angles of a polygon of 21 sides,


a21= 180° × (21 – 2) = 180° × 19 = 3420°


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