  We establish a relative trace formula on $\mathrm{GL}(n+1)$ weighted by cusp
forms on $\mathrm{GL}(n)$ over number fields. The spectral side is a weighted
average of Rankin-Selberg $L$-functions for
$\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over the \textit{full spectrum}, and the
geometric side consists of Rankin-Selberg $L$-functions for
$\mathrm{GL}(n)\times\mathrm{GL}(n),$ and certain explicit holomorphic
functions. The formula yields new results towards central $L$-values for
$\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ (over number fields): the second moment
evaluation, and simultaneous nonvnaishing in the level aspect.
