Interactive demonstration of mathematical induction for summation formulas showing how each step builds proof from n=1 through n=5

You already know the formula is true — induction just shows the mechanism

Most students think induction proves a formula from scratch. Actually, you verify it holds at the starting point, then demonstrate the algebraic machinery that carries truth forward one step at a time. Drag the slider to watch each case build on the previous one.

Proof Visualisation Base Case n=1
Each step assumes the previous case is true
Base: 1 = 1×2÷2 ✓
n=1 n=2 n=3 n=4 n=5
Left Side Sum
1
Formula Result
1
The base case is where you verify the formula works for the smallest value in your domain — typically n equals 1. For the sum 1 plus 2 plus 3 plus dot dot dot plus n equals n times the quantity n plus 1 divided by 2, you substitute n equals 1 into both sides: left side gives 1, right side gives 1 times 2 divided by 2 which equals 1. Both sides match, so the base case holds. This single verification is the foundation — without it, the entire inductive argument collapses because you need a valid starting point to build from.
Know This

Mathematical induction works like dominoes: the base case tips the first domino, and the inductive step ensures each fallen domino knocks over the next, proving the formula holds for all positive integers.