Relative Trace Formula and $L$-functions for $\mathrm{G}\mathrm{L}(n+1)\times \mathrm{G}\mathrm{L}(n)$

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