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Chapter 3: Problem 123
For the sequence \(r\) defined by $$r_{n}=3 \cdot 2^{n}-4 \cdot 5^{n}, \quad n \geq 0$$. Find \(r_{2}\).
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Let \(L\) be the set of all strings, including the null string, that can be constructed by repeated application of the following rules: If \(\alpha \in L,\) then \(a \alpha b \in L\) and \(b \alpha a \in L .\) If \(\alpha \in L\) and \(\beta \in L,\) then \(\alpha \beta \in L\) For example, \(a b\) is in \(L\), for if we take \(\alpha=\lambda,\) then \(\alpha \in L\) and the first rule states that \(a b=a \alpha b \in L\). Similarly, \(b a \in L\). As another example, aabb is in \(L\), for if we take \(\alpha=a b\), then \(\alpha \in L\); by the first rule, \(a a b b=a \alpha b \in L .\) As a final example, aabbba is in \(L\), for if we take \(\alpha=a a b b\) and \(\beta=b a,\) then \(\alpha \in L\) and \(\beta \in L ;\) by the second rule, \(a a b b b a=\alpha \beta \in L\). Show that baabab is in \(L\).
Let \(X\) be the set of positive integers that are not perfect squares. (A
perfect square \(m\) is an integer of the form \(m=i^{2}\) where \(i\) is an
integer.) Concern the sequence s from \(X\) to \(\mathbf{Z}\) defined as follows.
If \(n \in X,\) let \(s_{n}\) be the least integer \(a_{k}\) for which there exist
integers \(a_{1}, \ldots, a_{k}\) with \(n
List all strings over \(X=\\{0,1\\}\) of length 2 .
For the sequence \(r\) defined by $$r_{n}=3 \cdot 2^{n}-4 \cdot 5^{n}, \quad n \geq 0$$. Find \(r_{3}\)
If \(X\) and \(Y\) are sets, we define \(X\) to be equivalent to \(Y\) if there is a one-to-one, onto function from \(X\) to \(Y .\) Show that the sets \(\\{1,2, \ldots\\}\) and \(\\{2,4, \ldots\\}\) are equivalent.
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