If $α_{1}, α_{2}, α_{3}, ...... α_{n}$ are the $n^{t h}$ roots of unity, then $^{n} C_{1} . α_{1} +^{n} C_{2} . α_{2} + ....... +^{n} C_{n} . α_{n}$ is equal to $(H e r e α_{k} = e^{\frac{i 2 k π}{n}}; k = 1, 2, 3, ....., n) .$

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$\frac{α_{1}}{α_{2}} {{(α_{1} + α_{2})}^{2 n} - 1}$

$\frac{α_{2}}{α_{1}} {{(1 - α_{2})}^{2 n} - 1}$

$\frac{α_{1}}{α_{2}}$

$\frac{α_{1}}{α_{2}} {{(1 + α_{2})}^{n} - 1}$

answer is C.

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Detailed Solution

As $α_{1}, α_{2}, α_{3}, ...... α_{n}$ are the $n^{t h}$ roots of unity $\Rightarrow α_{1}, α_{2}, α_{3}, ...... α_{n}$ are in G.P

Where $α_{1} = 1, α_{2} = e^{i 2 π / n}, α_{3} = e^{i 4 π / n}, ..... α_{n} = e^{i 2 (n - 1) π / n}$

Clearly $α_{2}$ is common ratio and $α_{n} = α_{1} {(α_{2})}^{n - 1}$

Given exp $=^{n} C_{1} α_{1} +^{n} C_{2} α_{2} + .... +^{n} C_{n} α_{n} = α_{1} {^{n} C_{1} α_{1} +^{n} C_{2} α_{2} +^{n} C_{3} {(α_{2})}^{2} + ..... +^{n} C_{n} {(α_{2})}^{n - 1}}$

$= \frac{α_{1}}{α_{2}} {^{n} C_{1} α_{2} +^{n} C_{2} {(α_{2})}^{2} +^{n} C_{3} {(α_{2})}^{3} + ..... +^{n} C_{n} {(α_{2})}^{n}} = \frac{α_{1}}{α_{2}} {{(1 + α_{2})}^{n} - 1}$