Question

Assume that $A _{ i }( i =1,2, \ldots \ldots ., n )$ are the vertices of a regular polygon inscribed in a circle of radius unity then the value of $\displaystyle\sum_{i=1}^{n-1}\left|A_{1} A_{i+1}\right|^{2}$, is

• A. 2n
• B. n
• C. -n
• D. -2n

Answer 1

Answer: (A)

$\angle A _{1} OA _{ i +1}=\frac{2 i }{ n } \pi$
$\therefore A _{1} A _{ i +1}=2 OA _{1} \sin \frac{ i \pi}{ n }=2 \sin \frac{ i \pi}{ n }$
$\therefore \displaystyle\sum_{i=1}^{n-1}\left|A_{1} A_{i+1}\right|^{2}=4\left[\sin ^{2} \frac{\pi}{n}+\sin ^{2} \frac{2 \pi}{n}+\ldots .+\sin ^{2} \frac{(n-1) \pi}{n}\right]$
$=2\left[1-\cos \frac{2 \pi}{ n }+1-\cos \frac{4 \pi}{ n }+\ldots+1-\cos \frac{2( n -1) \pi}{ n }\right]$
$=2( n -1)-2\left[\left(1+\cos \frac{2 \pi}{ n }+\cos \frac{4 \pi}{ n }+\ldots\right)-1\right]$
$=2 n-2-2[0-1]=2 n$
$\left[\therefore 1+\cos \frac{2 \pi}{ n }+\cos \frac{4 \pi}{ n }+\ldots+\cos \frac{2( n -1) \pi}{2}=0 n ^{\text {th }}\right.$ roots of unity ]