Let \textrm{ }\textrm{ } \lambda = \underset{n \rightarrow \infty}{Lim} \textrm{ } \sum_{r = 1}^{n} \textrm{ }\textrm{ } \frac{1}{^{\textrm{ } r} C_{1} \cdot \left(\textrm{ }\right)^{r + 1} C_{1} \cdot \left(\textrm{ }\right)^{r + 2} C_{1}} . If one root of quadratic equation 16x2 – 12x + k = 0 is less than λ and other is greater than λ, then k can be

Engineering

Mathematics

Special Types of Sequence

Question

$Let λ = \underset{n \to \infty}{Lim} \sum_{r = 1}^{n} \frac{1}{^{r} C_{1} \cdot^{r + 1} C_{1} \cdot^{r + 2} C_{1}} .$ If one root of quadratic equation 16x² – 12x + k = 0 is less than λ and other is greater than λ, then k can be

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$∵ λ = \underset{n \to \infty}{Lim} \sum_{r = 1}^{n} \frac{1}{r (r + 1) (r + 2)}$

$= \underset{n \to \infty}{Lim} \frac{1}{2} \sum_{r = 1}^{n} (\frac{1}{r (r + 1)} - \frac{1}{(r + 1) (r + 2)})$

$= \underset{n \to \infty}{Lim} \frac{1}{2} (\frac{1}{1 \cdot 2} - \frac{1}{(n + 1) (n + 2)})$

$\Rightarrow λ = \frac{1}{4}$

$∵$ λ lies between the roots

$∴ 16 \cdot {(\frac{1}{4})}^{2} - 12 \cdot \frac{1}{4} + k < 0$

⇒ k < 2

$∴$ k can be –1, 0, 1