Question

The difference between 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.

Answer 1

The angles of a polygon will form A.P with common difference $d$ as $5$ and first term $a$ as $120$. It is known that sum of all angles of a polygon with $n$ sides is $180(n-1)$So, $S_n=180(n-2)$

We know,

$S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

Where,

$a=$ First term $=120$

$d=$ Common difference $=5$

$n=$ number of terms

$S_n=$ Sum of $n$ terms

$\Rightarrow \frac{n}{2}[2a+(n-1\left)d\right]=180(n-2)$
$\Rightarrow \frac{n}{2}[240+(n-1\left)5\right]=180(n-2)$

$\Rightarrow n=(240+(n-5\left)5\right)=360(n-2)$

$\Rightarrow 240n+5n^2-5n=360n-720$
$\Rightarrow 5n^2-125n+720=0$
$\Rightarrow n^2-25n+144=0$
$\Rightarrow n^2-10n-9n+144=0$
$\Rightarrow n(n-16)-9(n-16)=0$
$\Rightarrow (n-9)(n-16)=0$
$n=9$ or $n=16$