Get the complete, step-by-step math solution for: "How many ways can a 2×n rectangle be tiled using 1×2 dominoes?". Powered by SolveForX AI math tutor.
Step 1: Define the problem and base cases
Let an represent the number of ways to tile a 2×n rectangle using 1×2 dominoes. We need to find a recurrence relation for an. First, let's consider the base cases for small values of n.
Step 2: Calculate base cases for n=1 and n=2
For n=1, a 2×1 rectangle can only be tiled in one way: with a single vertical domino. So, a1=1. For n=2, a 2×2 rectangle can be tiled in two ways: either with two vertical dominoes or with two horizontal dominoes. So, a2=2.
Step 3: Derive the recurrence relation
Consider the rightmost column(s) of the 2×n rectangle. There are two possibilities for placing the last dominoes: \\ 1. A single vertical domino in the last column. The remaining 2×(n−1) rectangle can be tiled in an−1 ways. \\ 2. Two horizontal dominoes occupying the last two columns. The remaining 2×(n−2) rectangle can be tiled in an−2 ways. \\ These two cases are mutually exclusive and exhaustive, so an=an−1+an−2.
Step 4: Recognize the Fibonacci sequence
The recurrence relation an=an−1+an−2 with initial conditions a1=1 and a2=2 defines a shifted Fibonacci sequence. The standard Fibonacci sequence is F0=0,F1=1,F2=1,F3=2,F4=3,…. Comparing, we see that an=Fn+1.