<s>[INST] Prove the inequality $\,\frac{1}{\sqrt{1}+ \sqrt{3}} +\frac{1}{\sqrt{5}+ \sqrt{7} }+\ldots+\frac{1}{\sqrt{9997}+\sqrt{9999}}\gt 24$ 
Prove the inequality $$\frac{1}{\sqrt{1}+ \sqrt{3}}
 +\frac{1}{\sqrt{5}+ \sqrt{7} }+......... +\frac{1}{\sqrt{9997}+\sqrt{9999}} > 24$$

My work: 
Rationalizing the denominator gives 
$$\frac{\sqrt{3}-1}{2}  +\frac{\sqrt{7}-\sqrt{5}}{2}+......+\frac{\sqrt{9999}-\sqrt{9997}}{2} .$$
Now by taking two as common and separating the positive and negative terms gives
$$\frac{1}{2} [ \{\sqrt{3} +\sqrt{7}+\dots +\sqrt{9999}\} - \{1+\sqrt{5} +\dots+\sqrt{9997}\}].$$
Can we do like this please suggest. Thanks.
 [/INST] Note that $\frac{\sqrt{3}-1}{2}>\frac{\sqrt{5}-\sqrt{3}}{2}$, etc. because $\sqrt{\phantom{x}}$ is concave down. So twice your left-hand side is greater than a telescoping sum.
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