  We have derived effective gravitational field equations on a lower
dimensional hypersurface (known as a brane), placed in a higher dimensional
bulk spacetime for both Einstein and $f(\mathcal{R})$ gravity theories. We have
started our analysis on $n$-dimensional bulk from which the effective field
equations on a $(n-1)$-dimensional brane has been obtained by imposing $Z_{2}$
symmetry. Subsequently, we have arrived at the effective equations in
$(n-2)$-dimensions starting from the effective equations for $(n-1)$
dimensional brane. This analysis has been carried forward and is used to obtain
the effective field equations in $(n-m)$-dimensional brane, embedded in a
$n$-dimensional bulk. Having obtained the effective field equations in Einstein
gravity, we have subsequently generalized the effective field equation in
$(n-m)$-dimensional brane which is embedded the $n$-dimensional bulk spacetime
endowed with $f(\mathcal{R})$ gravity. We have also presented applications of
our results in the context of Einstein and $f(\mathcal{R})$ gravity. In both
the cases we have discussed vacuum static spherically symmetric solutions as
well as solutions in cosmological context. Implications are also discussed.
