Prove that (-1+i√3/2)^n +(-1-√3/2)^n is equal to (i) 2 if n is a multiple of 3 (ii) -1 if n - Maths - Complex Numbers and Quadratic Equations

Question

Prove that (-1+i√3/2)^n +(-1-√3/2)^n is equal to (i) 2
if n is a multiple of 3 (ii) -1 if n is not a multiple of 3

Brijendra Pal · Accepted Answer

HI,

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so here these are ω and ω2so our question becomes= ωn+ω2nnow if n is multiple of 3 then n = 3m= ω3m+ω2×3m=ω3m+ω32mand we know that ω3= 1 = 1+1= 2 now case 2 if n is not a multpile of 3 so it can be 3m+1 or 3m+2when n = 3m+1=ω3m+1+ω2×3m+1=ω3mω+ω32mω2=ω+ω2= -1 Since 1+ω2+ω=0again when n = 3m+2 =ω3m+2+ω2×3m+2=ω3mω2+ω32mω4=ω2+ω3ω=ω+ω2= -1 Since 1+ω2+ω=0 and ω3=1

Prove that (-1+i√3/2)^n +(-1-√3/2)^n is equal to (i) 2 if n is a multiple of 3 (ii) -1 if n is not a multiple of 3