Use strong induction to show that all dominoes fall in an in

Use strong induction to show that all dominoes fall in an infinite arrangement of dominoes if you know that the first three dominoes fall, and that when a domino falls, the domino three farther down in the arrangement also falls.

Solution

Let P(n) be the statement that the nth domino falls.

We need to prove that P(n) is true for all natural numbers n.

Notice, that the given condition holda for n=1,2,3.

Thus, P(1), P(2), andP(3) are true.

For the inductive step, put k?3 and

Let us suppose that P(j) is true for all j ? k.

We prove that P(k+1) is true.

Since k?3, there exist a natural numeber k-2 which is less than or may be equal to k, so by the
inductive hypothesis we know that P(k-2) is true.

So,we know that the (k-2)nd domino falls.
We have that