MATH SOLVE

4 months ago

Q:
# Let Dn denote the number of ways to cover the squares of a 2-by-n board using plain dominos. Then it is easy to see that D1 = 1, D2= 2, and D3 = 3. Compute a few more values of Dn and guess an expression for the value of Dn and use induction to prove you are right.

Accepted Solution

A:

Answer:See explanation below.Step-by-step explanation:The area of a plain domino equal the area of a 2-by-1 board, so it is easy to see that D4=4, D5=5, and so on.
Let's prove by induction than Dn=n for every natural number n.
1) Obviously, D1 is true.
2) Assume now that Dn is true for a given natural n, that is to say, a 2-by-n board can be covered with n dominoes.
we must prove that this implies D(n+1) is also true.
But the area of a 2-by-(n+1) board equals the area of a 2-by-n board plus the area of a 2-by-1 board. By 2), we can cover the 2-by-n board with n dominoes and by 1) we can cover the 2-by-1 board with 1 domino, so we just need n+1 dominoes to cover the 2-by-(n+1) board, which is what we wanted to prove.