The value of -n-2n- n- and - is a nonreal cube root of unity is

The correct options are A 1 if n is not a multiple of 3 C 2 if n is a multiple of 3 α^ n + α^ 2n = α^n + 1α^2n There are 2 cases, i) if n is a multiple o ...

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Standard XII

Mathematics

Cube Root of a Complex Number

The value of ...

Question

The value of $α^{- n} + α^{- 2 n}, n ϵ N$ and $α$ is a nonreal cube root of unity is

3 if n is a multiple of 3

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-1 if n is not a multiple of 3

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2 if n is a multiple of 3

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none of these

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Solution

The correct options are
A -1 if n is not a multiple of 3
C 2 if n is a multiple of 3
$α^{- n} + α^{- 2 n} = \frac{α^{n} + 1}{α^{2 n}}$
There are 2 cases, i) if n is a multiple of 3; $α^{n} = α^{2 n} = 1$
So, the answer comes out as $\frac{2 + 1}{1} = 2$ .
ii) If n is not a multiple of 3, $α^{n} + 1 = - α^{2 n}$ and so the answer becomes -1.

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The value of α−n+α−2n,nϵN and α is a nonreal cube root of unity is

The correct options are A -1 if n is not a multiple of 3 C 2 if n is a multiple of 3α−n+α−2n=αn+1α2nThere are 2 cases, i) if n is a multiple of 3; αn=α2n=1So, the answer comes out as 2+11=2.ii) If n is not a multiple of 3, αn+1=−α2n and so the answer becomes -1.

The value of $α^{- n} + α^{- 2 n}, n ϵ N$ and $α$ is a nonreal cube root of unity is