The interior angles of a polygon are in arithmetic progression. The smallest angle is `120^@` and the common difference is `5^@` Find the number of sides of the polygon

Explanation

Let 'n' be the number of sides of the polygon.
`:.` Sum of all the n interior angles of polygon=`(2n-4)xx90^(@) " " ...(1)`
and sum of 'n' angles of A.P. `=(n)/(2)[2xx120^(@)+(n-1)xx5^(@)]`
`=(n)/(2)(5n+235^(@) " " (2)`
Now, `(n)/(2)(5n+235)^(@)=(2n-4)xx90^(@) " " ` [from (1) and (2)]
`rArr 5n(n+47)=2(2n-4)xx90`
`rArr n^(2)+47n=72n-144`
`rArrn^(2)-25n+144=0`
`rArr (n-9)(n-16)=0`
`rArrn-9=0 or n-16=0`
`rArr n=9 or n=16`
Consider,`T_(9)=a+8d=120^(@)+8xx5^(@)=160^(@) lt 180^(@)`
and `T_(16)=a+15d=120^(@)+15xx5^(@)=195^(@) gt 180^(@)`
But each angle in a convex polygon will be less than `180^(@)`.
So, we reject `T_(16)`.
Hence, number of sides in a polygon = 9.