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Question

# State true or false:Assume that Ai(i=1,2,....n) are the vertices of a regular n-sided polygon inscribed in a circle of radius unity. Then,|A1A2|2+|A1A3|2+⋯+|A1An|2=−1. Type 1 for true and 0 for false

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Solution

## With origin as the centre of the circle of redius unity, let z1,z2,.....,zn represent the vertices A1,A2,....,An of the n-gon.Then we easily getz2=z1e2πi/n,z3=z1e4πi/n,.....,zn=1e2(n−1)πi/2nNow |A1Ar|2=|z1−zr|2=∣∣z1−z1e2(r−1)πi/n∣∣2=|z1|2∣∣1−e2(r−1)πi/n∣∣2.=∣∣∣1−cos2(r−1)πn−isin2(r−1)πn∣∣∣2[∵|z1|= radius of the circle=1]=(1−cos2(r−1)πn)2+sin22(r−1)πn=2−2cos2(r−1)πn ...(A)Hence n∑r=2|A1Ar|2=2(n−1)−2n∑r=2[cos2(r−1)πn] ...(1)Let S=n∑r=2cos2(r−1)πn ...(2)=cos2πn+cos4πn+cos6πn+⋯+⋯+cos2(n−1)nπFormula:cosA+cos(A+B)+cos(A+2B)+......+cos(A+¯¯¯¯¯¯¯¯¯¯¯¯¯n−1B)sinA+sin(A+B)+sin(A+2B)+....+sin(A+¯¯¯¯¯¯¯¯¯¯¯¯¯n−1B)There are n angles in A.P. of common diff. B.S=sinn.B2sinB2cos or sin{1stang.+lastang.2}In (2) there are (n-1) terms and angles are in A.P. of common difference 2πn.S=sin[(n−1).π/n]sin(π/n)cos⎧⎪⎨⎪⎩2πn+2(n−1)πn2⎫⎪⎬⎪⎭=sin(π−π/n)sin(π/n)cosπ=sin(π/n)sin(π/n)(−1)=−1Hence n∑r=2|A1Ar|2=2(n−1)−2S, by (1)=2n−2+2=2n as S=−1

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