Question

The interior angles of a convex polygon are in A.P. If the smallest angle is ${100}^{\circ }$ and the common difference is $4^{\circ }$, then the number of sides are

• A. $5$
• B. $7$
• C. $36$
• D. $44$

Answer 1

Answer: (A)

$5$

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 ${100}^0$ and common difference is $4^0$

$\therefore a=100andd=4$

Combining all the above relations and information, we get,

$\frac{n}{2}(2\times 100+(n-1)\times 4)=(n-2)\times 180$
$\Rightarrow n^2-41n+180=0$

Solving the above quadratic equation, we get,

$n=5$ or $n=36$

If we take $n=36$, then the biggest angle would be ${240}^0$, which is not possible in case of convex polygon.
Therefore, number of sides are 5.
Hence, option 'A' is correct.