Question - Recursive Sequence Calculation

u_0 = 2

u_{n+1} = 3u_n - 4n

u_1 = 3u_0 - 4 \cdot 0 = 3 \cdot 2 - 0 = 6

u_2 = 3u_1 - 4 \cdot 1 = 3 \cdot 6 - 4 = 18 - 4 = 14

u_3 = 3u_2 - 4 \cdot 2 = 3 \cdot 14 - 8 = 42 - 8 = 34

u_0 = 0

u_{n+1} = u_n^2 + \frac{1}{2n + 1}

u_1 = u_0^2 + \frac{1}{2 \cdot 0 + 1} = 0^2 + \frac{1}{1} = 0 + 1 = 1

u_2 = u_1^2 + \frac{1}{2 \cdot 1 + 1} = 1^2 + \frac{1}{3} = 1 + \frac{1}{3} = \frac{4}{3}

u_3 = u_2^2 + \frac{1}{2 \cdot 2 + 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}