For n≥2, let Sn denote the set of all subsets of {1,2,…,n} with no two consecutive numbers. For example, {1,3,5}∈S6, but {1,2,4}∈/S6. Then n(S5) is equal to _____
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Step-by-Step Solution
Step 1: Define the problem and notation
We are asked to find the number of such subsets for n=5. Let an=n(Sn) be the number of such subsets for a set of size n. We need to find a5.
Step 2: Calculate initial values for an
Let's start by finding the number of such subsets for small values of n. For n=0, there is one subset (the empty set). For n=1, the subsets are ∅ and {1}. For n=2, the subsets are ∅, {1}, {2}. Note that {1,2} is not allowed as 1 and 2 are consecutive.
Step 3: Derive a recurrence relation
Consider a subset A∈Sn. There are two cases: \\ Case 1: n∈/A. Then A must be a subset of {1,2,…,n−1} with no consecutive numbers. There are an−1 such subsets. \\ Case 2: n∈A. Then n−1∈/A (since n and n−1 are consecutive). So A must be of the form A′∪{n}, where A′ is a subset of {1,2,…,n−2} with no consecutive numbers. There are an−2 such subsets. \\ Thus, an=an−1+an−2. This is the Fibonacci sequence.
Step 4: Calculate a3,a4,a5
Using the recurrence relation an=an−1+an−2 and the initial values a1=2 and a2=3, we can calculate the subsequent terms. \\ a3=a2+a1=3+2=5. The subsets are ∅,{1},{2},{3},{1,3}. \\ a4=a3+a2=5+3=8. \\ a5=a4+a3=8+5=13.