Question

The interior angles of a polygon are in arithmetic progression. The smallest angle is ${120}^0$ and the common difference is $5^0$. The number of sides of the polygon is-

• A. $7$
• B. $9$
• C. $11$
• D. $16$

Answer 1

The correct option is B $9$

We know that, sum of $n$ terms of an AP, $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

We also know that, for an $n$ sided polygon, the sum of its interior angles is$:(n-2)\times {180}^0$

Given that, smallest angle is ${120}^0$ and common difference is $5^0$

$\therefore a=120andd=5$

Combining all the above relations and information, we get,

$\frac{n}{2}(2\times 120+(n-1)\times 5)=(n-2)\times 180$
$\therefore n^2-25n+144=0$
$\therefore n=9,16$
But if $n=16$ then $T_{16}={195}^o$, which is not possible
$\therefore n=9$.