Find asymptotics for solution $x$ of $(x+1)^{\frac{n+1}{n}}-x^{\frac{n+1}{n}}=5$

Question

It is easy to see that for any $n\geq 1$, the equation $(x+1)^{\frac{n+1}{n}}-x^{\frac{n+1}{n}}=5$ has a unique positive solution ; call it $x_n$.

Is there a simple asymptotic formula for $x_n$ ? I tried unsuccessfully to find one, and computed that

$$ \lfloor x_3 \rfloor=53, \ \lfloor x_4 \rfloor=256, \ \lfloor x_5 \rfloor=1256, \ \lfloor x_6 \rfloor=6195, \ \lfloor x_7 \rfloor=30678, \ \lfloor x_8 \rfloor=152243, \ $$

It seems that the sequence $(\frac{x_{n+1}}{x_n})$ is increasing and converges to $5$.

Answer 1

By the mean value theorem, we have

$$5 = (x_n+1)^{(n+1)/n} - x_n^{(n+1)/n} = \frac{n+1}{n}(x_n + \xi_n)^{1/n}\cdot 1$$

for some $\xi_n \in (0,1)$. That yields

$$x_n + \xi_n = \frac{5^n}{\left(1+\frac1n\right)^n} = \frac{5^n}{e}\exp\left(\frac{1}{2n} + O\left(n^{-2}\right)\right).$$