If n is an odd positive integer and I- 1- 2- n-1 are the n-nth roots of unity- then 3- 1 3- 2 - 3- n-1 equals

The correct option is C 3^n+1/4 x^n 1=(x 1)(x α_1)(x α_2)(x α_3)...(x α_n 1) x^n 1x 1=(x α_1)(x α_2)(x α_3)...(x α_n 1) (x α_1)(x α_2)(x α_ ...

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Standard XII

Mathematics

Properties of nth Root of a Complex Number

If n is an od...

Question

If n is an odd positive integer and $I, α_{1}, α_{2}, . . . . α_{n - 1}$ are the $n, n^{t h}$ roots of unity, then $(3 + α^{1}) (3 + α^{2}) . . . . (3 + α^{n - 1})$ equals

\frac{3^{n} + 1}{4}

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\frac{3^{n} - 1}{2}

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\frac{3^{n} - 1}{4}

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None of these

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Solution

The correct option is C $\frac{3^{n} + 1}{4}$
$x^{n} - 1 = (x - 1) (x - α_{1}) (x - α_{2}) (x - α_{3}) . . . (x - α_{n - 1})$
$\frac{x^{n} - 1}{x - 1} = (x - α_{1}) (x - α_{2}) (x - α_{3}) . . . (x - α_{n - 1})$
$(x - α_{1}) (x - α_{2}) (x - α_{3}) . . . (x - α_{n - 1}) = 1 + x + x^{2} + . . . x^{n - 1}$
Substituting $x = - 3$ .
$(3 + α_{1}) (3 + α_{2}) (3 + α_{3}) . . . (3 + α_{n - 1}) = 1 - 3 + 3^{2} + . . . (- 1)^{n - 1} 3^{n - 1}$
Therefore $(3 + α_{1}) (3 + α_{2}) (3 + α_{3}) . . . (3 + α_{n - 1})$
$= \frac{1 - (- 3)^{n}}{1 - (- 3)}$
Now, $n$ is odd, therefore
$= \frac{3^{n} + 1}{4}$

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If n is an odd positive integer and I,α1,α2,....αn−1 are the n,nth roots of unity, then (3+α1)(3+α2)....(3+αn−1) equals

If n is an odd positive integer and $I, α_{1}, α_{2}, . . . . α_{n - 1}$ are the $n, n^{t h}$ roots of unity, then $(3 + α^{1}) (3 + α^{2}) . . . . (3 + α^{n - 1})$ equals