How to solve this particular recurrence?

Question

$x_{n+1}=((n+1)/n)x_{n}+x_n+n+1, x_1=5$

I have calculated $x_2= 17, x_3= 35.5, $

But I don't know how to go forward.

Please help.

Answer 1

I suppose that there is a typo and that more probably the equation could be $$x_{n+1}=\frac{n+1}nx_{n}+n+1$$ and not what you wrote (which could be extremely difficult to solve.

Define $y_n=\frac{x_n}n$ to get $$y_{n+1}=y_n+1$$ which would lead to $y_n=n+C$ that is to say $$x_{n}=n(n+C)$$ and the condition would give $C=4$ making $$x_n=n(n+4)$$

Edit

Just to give you an idea of the complexity, if it was (as written) $$x_{n+1}=\frac{n+1}nx_{n}+\color{red}{x_n}+n+1$$ with $x_1=5$, the solution would be $$x_n=\frac{\left(\frac{3}{2}\right)_{n-1} \left((10+\pi ) 2^n \Gamma \left(n+\frac{3}{2}\right)-\sqrt{\pi } \, _2F_1\left(1,n+2;n+\frac{3}{2};\frac{1}{2}\right) \Gamma (n+2)\right)}{4 \Gamma (n) \Gamma \left(n+\frac{3}{2}\right)}$$ where appear Pochhammer symbols, gamma function and hypergeometric function. Quite nice, isn't it ?