Question : There are two regular polygons with numbers of sides equal to $(n-1)$ and $(n+2)$. Their exterior angles differ by 6°. The value of $n$ is:
Option 1: 14
Option 2: 12
Option 3: 13
Option 4: 11
Correct Answer: 13
Solution :
Given: There are two regular polygons with numbers of sides equal to ($n$–1) and ($n$+2). Their exterior angles differ by 6°.
We know that exterior angle of a polygon = $\frac{360°}{\text{number of sides}}$
According to the question,
$\frac{360°}{n-1}-\frac{360°}{n+2}=6°$
$⇒360°(\frac{1}{n-1}-\frac{1}{n+2})=6°$
$⇒60°[\frac{n+2-n+1}{(n-1)(n+2)}]=1$
$⇒n^2+n-2=180$
$⇒n^2+n-182=0$
$⇒(n+14)(n-13)=0$
$\therefore n=13$ (Since $n$ can't be negative)
Hence, the correct answer is 13.
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