  We prove a new case of spectral reciprocity formulae for the product of
$GL(n+1) \times GL(n)$ and $GL(n) \times GL(n-1)$ Rankin-Selberg $L$-functions
($n \geq 3$), which are first developed by Blomer and Khan in \cite{BK17} for
degree 8 $L$-functions ($n=2$ case, the product of $GL(3) \times GL(2)$ and
$GL(2) \times GL(1)$ Rankin-Selberg $L$-functions). Our result can be viewed as
a generalization of Blomer and Khan's work to higher rank case. We will mainly
follow the method developed in \cite{Nun20}. We will use the integral
representations of Rankin-Selberg $L$-functions generalized by Ichino and
Yamana \cite{IY15}, spectral theory of $L^2$ space and the language of
automorphic representations.
