Chapter 31 - Atomic Physics - Problems and Conceptual Exercises - Page 1112: 19

$(\frac{n}{n+1})^2U$

We know that $U_n=\frac{-KZe^2}{r_n}$ $\implies U_n\propto \frac{1}{r_n}$ As $r_n\propto n^2$ $\implies U_n\propto \frac{1}{n^2}$ We know that $U_n\times n^2=constant$ Thus, $(U_{n+1})(n+1)^2=(U_n)(n^2)$ $\implies U_{n+1}=U_n[\frac{n^2}{(n+1)^2}]$ $\implies U_{n+1}=(\frac{n}{n+1})^2U$