Question

If b₁, b₂,…..b_n are the n^th roots of unity, then ⁿC₁. b₁ + ⁿC₂.b₂ +………+ ⁿC_n.b_n is equal to

• A. $\frac{b_{1}}{b_{2}}$
• B. $\frac{b_{1}}{b_{2}}\left\{ (b_{1} + b_{2})^{2n} - 1 \right\}$
• C. $\frac{b_{1}}{b_{2}}\left\{ (1 + b_{2})^{n} - 1 \right\}$
• D. None of these

Answer 1

Answer: (C)

$\frac{b_{1}}{b_{2}}\left\{ (1 + b_{2})^{n} - 1 \right\}$

Answer 2

As b₁, b₂,…..b_n are n^th roots of unity

Ž b₁,b₂,……b_n are in G.P.

where b₁ = 1, b₂ = e^i2p/n, b₃ = e^i4p/n….

b_n = $e^{\frac{2i(n - 1)\pi}{n}}$

Clearly b₂ is common ratio and b_n = b₁(b₂)^n–1

given expression = ⁿC₁. b₁ + ⁿC₂. b₂ +…+ ⁿC_n . b_n

= b₁ (ⁿC₁ + ⁿC₂b₂ + ⁿC₂ b₂² +….+ ⁿC_n b₂^n–1)

= $\frac{b_{1}}{b_{2}}$ ((1 + b₂)ⁿ – 1)