First slide

Binomial theorem for positive integral Index

Question

$I f C (n - 1, r) = (k^{2} - 3) \cdot C (n, r + 1) t h e n k \in$

Moderate

A

$(- \infty, - 2]$
B

$[2, \infty)$
C

$[\sqrt{3}, 2]$
D

$[- 2, - \sqrt{3}] \cup [\sqrt{3}, 2]$

Solution

$\begin{array}{l} Given C (n - 1, r) = (k^{2} - 3) (C (n, r + 1)) \\ it gives \\ \frac{(n - 1)!}{r! (n - r - 1)!} = (k^{2} - 3) \frac{n!}{(r + 1)! (n - r - 1)!} \\ k^{2} - 3 = \frac{r + 1}{n} \\ 0 < k^{2} - 3 < 1 \\ 3 < k^{2} < 4 \end{array}$
$Therefore, k \in [- 2, - \sqrt{3}] \cup [\sqrt{3}, 2]$

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