[JEE Main 2015] For integer $k$, $\alpha_k=\cos\frac{k\pi}7+i\sin\frac{k\pi}7$. The value of $\dfrac{\sum_{k=1}^{12}|\alpha_{k+1}-\alpha_k|}{\sum_{k=1}^{3}|\alpha_{4k-1}-\alpha_{4k-2}|}$ is ____
For integer $k$, $\alpha_k=\cos\frac{k\pi}7+i\sin\frac{k\pi}7$. The value of $\dfrac{\sum_{k=1}^{12}|\alpha_{k+1}-\alpha_k|}{\sum_{k=1}^{3}|\alpha_{4k-1}-\alpha_{4k-2}|}$ is ____
1 Answer
VAVidaara Admin
✓ Vidaara Team
✓ Accepted
· 1mo ago
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Answer: $4$.
Each $|\alpha_{k+1}-\alpha_k|=|e^{i\pi/7}-1|=2\sin\frac\pi{14}$ — a constant, since consecutive indices differ by $1$. Numerator $=12c$, denominator $=3c$, ratio $=4$.
JEE Main 2015 · Complex Numbers — verified solution by the Vidaara Team.
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