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Sum of N Terms of an AP

If ω is fif...

Question

If $ω$ is fifth root of unity and $E = {(1 + 1)}^{n} + {(1 + ω)}^{n} + {(1 + ω^{2})}^{n} + {(1 + ω^{3})}^{n} + {(1 + ω^{4})}^{n}$ then

A

E is real

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B

E is

5 [C_{0} + C_{5} + C_{10} + C_{15} . . . . . .]

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C

E is

5 [C_{1} + C_{6} + C_{11} + . . . .]

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D

E is Non real

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Solution

The correct options are
A E is real
B E is $5 [C_{0} + C_{5} + C_{10} + C_{15} . . . . . .]$

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Similar questions

z

as real or complex such that

{(1 + z^{2} + z^{4})}^{8} = C_{0} + C_{1} z^{2} + C_{2} z^{4} \cdot \cdot \cdot \cdot \cdot \cdot + C_{16} z^{32}

C_{0} + C_{3} + . . . . + C_{15} + (C_{2} + C_{5} + C_{8} + C_{11} + C_{14}) ω +

(C_{1} + C_{4} + C_{7} + . . . . + C_{16}) ω^{2}

ω

is the cube root of unity .

z

is non zero complex number and

(1 + z^{2} + z^{4})^{8} = C_{0} + C_{1} z^{2} + C_{2} z^{4} + . . . + C_{16} z^{32}

ω

is a non real cube root of unity and n is a positive integer which is not a multiple of 3, then

1 + ω^{n} + ω^{2 n}

ω (\neq 1)

is a cube root of unity and

(1 + ω^{2})^{n} = (1 + ω^{4})^{n}

, then the least positive value of n is

1, ω, ω^{2}, . . ., ω^{n - 1}

n^{t h}

roots of unity, then the value of

(5 - ω) (5 - ω^{2}) . . . (5 - ω^{n - 1}) =

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