You are given the sequence: $$\frac{1}{4}, 1, \frac{9}{4}, 2 + \frac{2}{\sqrt{3}}, \frac{5}{2} + \frac{5}{2 \sqrt{2}}, 3 + 3 \sqrt{\frac{1}{10} (5 + \sqrt{5})}$$
What is the next number? Explain
Hint: The next number is rational.
This is my own puzzle
I'm not sure, if this is the intended one. But from $9/4$ onwards we can write: $$\small{\begin{align}& \frac{9}{4}, 2 + \frac{2}{\sqrt{3}}, \frac{5}{2} + \frac{5}{2 \sqrt{2}}, 3 + 3 \sqrt{\frac{1}{10} (5 + \sqrt{5})}\\ \Rightarrow&\frac{3}{2}\left(1+\frac{1}{2\left(1\right)}\right),\frac{4}{2}\left(1+\frac{1}{2\left(\frac{\sqrt{3}}{2}\right)}\right),\frac{5}{2}\left(1+\frac{1}{2\left(\frac{1}{\sqrt{2}}\right)}\right),\frac{6}{2}\left(1+\frac{1}{2\left(\sqrt{\frac{5-\sqrt{5}}{8}}\right)}\right) \\ \Rightarrow& \frac{3}{2}\left(1+\frac{1}{2}\csc\left(\frac{\pi}{2}\right)\right),\frac{4}{2}\left(1+\frac{1}{2}\csc\left(\frac{\pi}{3}\right)\right),\frac{5}{2}\left(1+\frac{1}{2}\csc\left(\frac{\pi}{4}\right)\right),\frac{6}{2}\left(1+\frac{1}{2}\csc\left(\frac{\pi}{5}\right)\right)\end{align}}$$
If this is the pattern, then the next term will be: $$\frac{7}{2}\left(1+\frac{1}{2}\csc\left(\frac{\pi}{6}\right)\right)=7$$
Which is also rational. I'll update on this later.