The difference between any two consecutive interior angles of a polygon is $5^{\circ}$ If the smallest angle is $120^{\circ},$ find the number of the sides of the polygon.

Vedclass pdf generator app on play store
Vedclass iOS app on app store

The angles of the polygon will form an $A.P.$ with common difference $d$ as $5^{\circ}$ and first term $a$ as $120^{\circ}$

It is known that the sum of all angles of a polygon with $n$ sides is $180(n-2)$

$\therefore S_{n}=180^{\circ}(n-2)$

$\Rightarrow \frac{n}{2}[2 a+(n-1) d]=180^{\circ}(n-2)$

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

$\Rightarrow n[240+(n-1) 5]=360(n-2)$

$\Rightarrow 240 n+5 n^{2}-5 n=360 n-720$

$\Rightarrow 5 n^{2}-125 n+720=0$

$\Rightarrow n^{2}-25 n+144=0$

$\Rightarrow n^{2}-16 n-9 n+144=0$

$\Rightarrow n(n-16)-9(n-16)=0$

$\Rightarrow(n-9)(n-16)=0$

$\Rightarrow n=9$ or $16$

Similar Questions

Let $a, b, c, d, e$ be natural numbers in an arithmetic progression such that $a+b+c+d+e$ is the cube of an integer and $b+c+d$ is square of an integer. The least possible value of the number of digits of $c$ is

  • [KVPY 2013]

In an $A.P.,$ the first term is $2$ and the sum of the first five terms is one-fourth of the next five terms. Show that $20^{th}$ term is $-112$

If the sum of first $11$ terms of an $A.P.$, $a_{1} a_{2}, a_{3}, \ldots$is $0\left(\mathrm{a}_{1} \neq 0\right),$ then the sum of the $A.P.$, $a_{1}, a_{3}, a_{5}, \ldots, a_{23}$ is $k a_{1},$ where $k$ is equal to 

  • [JEE MAIN 2020]

If $n$ arithmetic means are inserted between a and $100$ such that the ratio of the first mean to the last mean is $1: 7$ and $a+n=33$, then the value of $n$ is

  • [JEE MAIN 2022]

If the sum of $n$ terms of an $A.P$. is $2{n^2} + 5n$, then the ${n^{th}}$ term will be