Complex Numbers
Complex Numbers
Allen Star Batch
Grade 11
Question:
If $z_n = \cos\frac{\pi}{n(n+1)(n+2)} + i\sin\frac{\pi}{n(n+1)(n+2)}$ for $n = 1, 2, 3, \ldots, k$, then the value of $\lim_{k\to\infty}\left|z_1z_2\cdots z_k\right|$ is:
$1$
$2$
$-\sum_{k=1}^{98}e^{\frac{2\pi k}{99}}$
$\sum_{k=0}^{98}e^{\frac{2\pi k}{99}}$
Step-by-Step Solution
Key Concept: Verify solutions by substitution into constraint equations, then count valid selections using binomial coefficients.
We verify that $0^2 + 0^2 = 0$, $1^2 + 2^2 = 5$, $2^2 + 4^2 = 20$, and $3^2 + 4^2 = 25$ satisfy the constraint equations. The required number of ways is ${}^9C_1 = 9$, and by symmetry ${}^9C_5 = {}^9C_{9-5} = {}^9C_4 = 9$.
Correct Answer: 1,3