  Let $n\in {\Bbb N}$. An element $a\in R$ has generalized n-strongly Drazin
inverse if there exists $x\in R$ such that $xax=x, x\in comm^2(a), a^n-ax\in
R^{qnil}.$ For any $a,b\in R$, we prove that $1-ab$ has generalized n-strongly
Drazin inverse if and only if $1-ba$ has generalized n-strongly Drazin inverse.
Extensions in Banach algebra are also obtained.
