Question - Recursion Sequences Calculation

\( u_0 = 2 \)

\( u_1 = 3u_0 - 4\cdot0 = 3\cdot2 - 0 = 6 \)

\( u_2 = 3u_1 - 4\cdot1 = 3\cdot6 - 4 = 18 - 4 = 14 \)

\( u_3 = 3u_2 - 4\cdot2 = 3\cdot14 - 8 = 42 - 8 = 34 \)

\( u_0 = 0 \)

\( u_1 = u_0^2 + \frac{1}{2\cdot0 + 1} = 0 + 1 = 1 \)

\( u_2 = u_1^2 + \frac{1}{2\cdot1 + 1} = 1 + \frac{1}{3} = \frac{4}{3} \)

\( u_3 = u_2^2 + \frac{1}{2\cdot2 + 1} = \left(\frac{4}{3}\right)^2 + \frac{1}{5} = \frac{16}{9} + \frac{1}{5} = \frac{80}{45} + \frac{9}{45} = \frac{89}{45} \)