Sequences & Series
Number of Sides of Polygon from AP Angles
nta_pyq_2025_apr
Grade 11
Question:
The interior angles of a polygon are in AP with common difference $6°$. If the largest interior angle is $219°$, then the number of sides $n$ of the polygon is
Step-by-Step Solution
Key Concept: Express the smallest angle $a_1$ from the largest angle and use the polygon angle-sum formula $(n-2)\times180°$ to form a quadratic in $n$.
Largest angle: $a_1+(n-1)\cdot6=219\Rightarrow a_1=225-6n$.
Sum of interior angles: $\frac{n}{2}(a_1+219)=(n-2)\cdot180$.
$\frac{n}{2}(225-6n+219)=(n-2)\cdot180$
$n(222-3n)=180(n-2)$
$222n-3n^2=180n-360$
$3n^2-42n-360=0\Rightarrow n^2-14n-120=0\Rightarrow(n-20)(n+6)=0$.
$n=20$ (taking the positive root).
Correct Answer: 20