Several concise formulations of mathematical induction are presented and proven equivalent. The formulations are expressed in variable-free relation algebra and thus are in terms of relations only, without mentioning the related objects. It is shown that the induction principle in this form, when combined with the explicit use of Galois connections, lends itself very well for use in calculational proofs. Two non-trivial examples are presented. The first is a proof of a Newman's lemma. The second is a calculation of a condition under which the union of two well-founded relations is wellfounded. In both cases the calculations lead to generalisations of the known results. In the case of the latter example, one lemma generalises three different conditions.