Question

the interior angles of an polygon are in A.P. the smallest angle is 120° and the common difference is 5 .Find the number of sides of the polygon

Answer 1

hello,
given that a=120° and d=5°
we know that S=n/2(2a+(n-1)d)
here S=(n-2)180°
we have to find n
(n-2)180=n/2(2·120+(n-1)5)
(n-2)360=n(240+5n-5)
(n-2)360=235n+5n²
360n-720=235n-5n²
5n²-125n+720=0
5(n²-25n+144)=0
n²-25n+144n=0
n²-16n-9n+144=0
n(n-16)-9(n-16)=0
(n-9)(n-16)=0
n=9 or n=16
but n≠16 
as 16th term =120+(16-1)5
=195° but this more than 180°
hence n=9
therefore,no. of sides is 9