What is Proof by Induction?
- The best way to visualize proof by induction is to visualize a row of dominoes.
- If one is knocked over, we assume that the next one will topple over.
- This starting point is called the basic step.
- We can easily prove this by putting two dominos next to each other and knocking one over.
- Then we assume that the domino in the kth place falls if the domino before it, the (k-1)th domino falls as well.
- This is the second step of the process.
- If you were to add another domino, the (k+1)th domino, then based on your initial proof and assumption in the first and second steps, you can assume that the (k+1)th domino will also fall.
- This is the third step of the process, called the inductive step.
- You can then finalize your argument by stating that at the start of the process it was shown that the first domino caused a second domino to fall.
- You can now use the second and third step to show that a (k+2)th domino will also fall over.
- As k is a variable, you could repeat topple as many dominoes as you want to prove that it's true, therefore the inductive proof should show that a domino will fall over as long as the one before it falls over, for any value k of dominoes.
Example
Sources
https://www.snexplores.org/article/falling-dominoes-speed-friction-physics