With induction, one uses a very elementary examplecomparatively abstract understanding of how a property propogates from smaller instances to larger instances, we in order to arrive at a fuller understanding of the property in particular cases, without need for explicit calculation. Thus, one can see that a particular finite graph or group or whatever kind of structure has a property, not by calculating it in that instance, but by an abstract inductive argument, on size or degree or rank or whatever. A complex graph-theoretic calculation is avoided by understanding what happens in general when a point is deleted.
And there are, of course, extremely concrete elementary instances. We all know, for example, how to usevalues for concrete sums $1+2+\cdots+105$. And surely mathematics is covered with dozens or hundreds of similar examplesSimilarly, of every degree of complexity. We we often understand the iterates of a function $f^n(x)$ without calculating them, or the powers of a matrix $A^n$, or the successive derivatives of a function, all without calculation, by understanding the inductive relationship in effect at each step.
The same phenomenon
Surely mathematics is exhibited in more abstract applications of induction. To prove a property of all finite graphs, all finite groups covered with dozens or whatever kind hundreds of structure you have, by induction on the sizesimilar examples, or of every degree or rank or whatever you are inducting on, replaces a calculation of the property in the individual cases with an abstract understanding complexity and every level of how the property propogates to more complex objectsabstraction.