$$1+ \frac{ 2 }{ \frac{ 3 }{ 2+ \frac{ 1 }{ 1+ \frac{ 1 }{ 2 } } } }$$
$\frac{25}{9}\approx 2.777777778$
$$1+\frac{2\left(2+\frac{1}{1+\frac{1}{2}}\right)}{3}$$
$$1+\frac{2\left(2+\frac{1}{\frac{2}{2}+\frac{1}{2}}\right)}{3}$$
$$1+\frac{2\left(2+\frac{1}{\frac{2+1}{2}}\right)}{3}$$
$$1+\frac{2\left(2+\frac{1}{\frac{3}{2}}\right)}{3}$$
$$1+\frac{2\left(2+1\times \frac{2}{3}\right)}{3}$$
$$1+\frac{2\left(2+\frac{2}{3}\right)}{3}$$
$$1+\frac{2\left(\frac{6}{3}+\frac{2}{3}\right)}{3}$$
$$1+\frac{2\times \frac{6+2}{3}}{3}$$
$$1+\frac{2\times \frac{8}{3}}{3}$$
$$1+\frac{\frac{2\times 8}{3}}{3}$$
$$1+\frac{\frac{16}{3}}{3}$$
$$1+\frac{16}{3\times 3}$$
$$1+\frac{16}{9}$$
$$\frac{9}{9}+\frac{16}{9}$$
$$\frac{9+16}{9}$$
$$\frac{25}{9}$$
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$\frac{5 ^ {2}}{3 ^ {2}} = 2\frac{7}{9} = 2.7777777777777777$
