What's the general term of the squence $(U_n)_n$ such that $U_n+\frac{1}{U_{n-1}}=2$?

Question

$$ U_n+\frac{1}{U_{n-1}}=2 $$ How we can find the general term of this sequence?

Answer 1

After you have properly specified the initial condition $U_0$ (for instance, one cannot have $U_0=0$; and $U_0 = 1$ is trivial), a simple approach is looking at a few terms and seeing what happens.

$$\begin{align} U_1 &= 2-\frac{1}{U_0} \\ U_2 &= 2-\frac{1}{U_1} = 2-\frac{1}{2-\frac{1}{U_0}}=\frac{3-\frac{2}{U_0}}{2-\frac{1}{U_0}} \\ U_3 &= 2-\frac{1}{U_2} = 2-\frac{2-\frac{1}{U_0}}{3-\frac{2}{U_0}}=\frac{4-\frac{3}{U_0}}{3-\frac{2}{U_0}} \end{align} $$ etc.

This will give you an idea of what to prove. Namely, one could guess that the general form is $$ U_n = \frac{n+1-\frac{n}{U_0}}{n-\frac{n-1}{U_0}} $$ and then proceed to prove it by induction.