Interactive picture of proof by induction for the sum 1 + 2 + … + n = n(n+1)/2, with a base case view that substitutes n = 1 and an inductive step view where the term k + 1 joins the assumed sum

Checking n = 1 to 5 proves five cases. One line of algebra proves them all

Testing a formula on a handful of values is not a proof: the next value could still break it. Induction verifies n = 1 once, then takes an unspecified k, assumes the formula there, and shows the next term carries it to k + 1. Switch to the inductive step and watch the term k + 1 join the assumed sum; then change k and notice the algebra never changes.