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We have derived effective gravitational field equations on a lowerdimensional hypersurface (known as a brane), placed in a higher dimensionalbulk spacetime for both Einstein and $f(\mathcal{R})$ gravity theories. We havestarted our analysis on $n$-dimensional bulk from which the effective fieldequations 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 obtainthe effective field equations in $(n-m)$-dimensional brane, embedded in a$n$-dimensional bulk. Having obtained the effective field equations in Einsteingravity, we have subsequently generalized the effective field equation in$(n-m)$-dimensional brane which is embedded the $n$-dimensional bulk spacetimeendowed with $f(\mathcal{R})$ gravity. We have also presented applications ofour results in the context of Einstein and $f(\mathcal{R})$ gravity. In boththe cases we have discussed vacuum static spherically symmetric solutions aswell as solutions in cosmological context. Implications are also discussed.

Effective field equation on m-brane embedded in n-dimensional bulk of Einstein and f(R) gravity