Let $a_{k} = \frac{k}{(k - l)^{4/3} + k^{4/3} + (k + l)^{4/3}}$ and $S_{n} = \sum_{k = 1}^{n}\mspace

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Let $a_{k} = \frac{k}{(k - l)^{4/3} + k^{4/3} + (k + l)^{4/3}}$ and $S_{n} = \sum_{k = 1}^{n}\mspace{2mu} a_{k}$, then

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Sol. [Key Concept: Sequence relations from the definition.]$$\frac{k}{(k - 1)^{4/3} + k^{4/3} + (k + 1)^{4/3}} < \frac{k}{(k - 1)^{4/3} + ((k - 1)(k + 1))^{2/3} + (k + 1)^{4/3}}$$$$\begin{matrix} & \ = \frac{\left( (k + 1)^{2/3} - (k - 1)^{2/3} ight)k}{(k + 1)^{2} - (k - 1)^{2}} \ & \ = \frac{1}{4}\left( (k + 1)^{2/3} - (k - 1)^{2/3} ight) \end{matrix}$$$$\begin{matrix} \Rightarrow & & \ \ \ \ \ \ \ \ & S_{n} & \ < \frac{1}{4}\sum_{k = 1}^{n}\mspace{2mu}\mspace{2mu}\left( (k + 1)^{2/3} - (k - 1)^{2/3} ight) = \frac{1}{4}\left( (n + 1)^{2/3} + n^{2/3} - 1 ight) \ \Rightarrow & & \ \ \ \ \ \ \ \ & S_{999} & \ < \frac{1}{4}\left( (1000)^{2/3} + (999)^{2/3} - 1 ight) < \frac{1}{4}(100 + 100 - 1) < 50 \ & & \ \ \ \ \ \ \ \ & {	ext{ }S}_{26} & \ < \frac{1}{4}\left( (27)^{2/3} + (26)^{2/3} - 1 ight) < \frac{1}{4}(9 + 9 - 1) = \frac{17}{4} \end{matrix}$$✘ (a) $S_{26} < 174$✔ (b) $S_{26} < \frac{14}{big}{((27)}^{2/3}{+ (26)}^{2/3} - 1big) < 174$✔ (c) $S_{999} < 50$✘ (d) $S_{999} < 50$

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