Question

Assume that $A_i(i=1,2,\dots \dots .,n)$ are the vertices of a regular polygon inscribed in a circle of radius unity then the value of $\sum _{i=1}^{n-1}{\left|A_1{A}_{i+1}\right|}^2$, is

• A. 2n
• B. n
• C. -n
• D. -2n

Answer 1

Answer: (A)

$\angle A_{1}O{A}_{i+1}=\frac{2i}{n}\pi$
$\therefore A_1{A}_{i+1}=2O{A}_1\sin \frac{i\pi }{n}=2\sin \frac{i\pi }{n}$
$\therefore \sum _{i=1}^{n-1}{\left|A_1{A}_{i+1}\right|}^2=4\left[{\sin }^2\frac{\pi }{n}+{\sin }^2\frac{2\pi }{n}+\dots .+{\sin }^2\frac{(n-1)\pi }{n}\right]$
$=2\left[1-\cos \frac{2\pi }{n}+1-\cos \frac{4\pi }{n}+\dots +1-\cos \frac{2(n-1)\pi }{n}\right]$
$=2(n-1)-2\left[\left(1+\cos \frac{2\pi }{n}+\cos \frac{4\pi }{n}+\dots \right)-1\right]$
$=2n-2-2[0-1]=2n$
$[\therefore 1+\cos \frac{2\pi }{n}+\cos \frac{4\pi }{n}+\dots +\cos \frac{2(n-1)\pi }{2}=0n^{\text{th }}$ roots of unity ]