If ( 1 ) be a cube root of unity and ( 1 + 2 ) n = ( 1 + 4 ) n , then the least positive value of n is [2004]

Question

If $ω (\neq 1)$ be a cube root of unity and ${(1 + ω^{2})}^{n} = {(1 + ω^{4})}^{n}$ , then the least positive value of $n$ is [2004]

Smart Achievers · Accepted Answer

(2)

${(1 + ω^{2})}^{n} = {(1 + ω^{4})}^{n}$

$\Rightarrow {(- ω)}^{n} = {(1 + ω)}^{n} = {(- ω^{2})}^{n}$

$\Rightarrow ω^{n} = 1 \Rightarrow n = 3$

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