Problem 1
Show that for \(n=0,1,2\) the following is true: $$1^{2}+2^{2}+3^{2}+\cdots+n^{2}=n(n+1)(2 n+1) / 6$$
Problem 1
The terms of a sequence are given recursively as \(a_{0}=2, a_{1}=6,\) and \(a_{n}=2 a_{n-1}+\) \(3 a_{n-2}\) for \(n \geq 2\). Find the first eight terms of this sequence.
Problem 1
Let \(A=\\{1,2,3, \ldots, 10\\}, B=12,3,6,8\\},\) and \(C=(3,5,4,8,2\\} .\) Find the following: (a) \(B \cup C\) (b) \(B \cap C\) (c) \(B-C\) (d) \(A-B\) (e) \(A-C\)
Problem 1
In a class of 35 students who are either biology majors or have blonde hair, there are 27 biology majors and 21 blondes. How many biology majors must be blonde?
Problem 1
Translate the following expressions into propositional logic. Use the following proposition letters: \(p="\) Jones told the truth." \(q={ }^{*}\) The butler did it." \(r=" I^{\prime} \|\) eat my hat." \(s=\) "The moon is made of green cheese." \(t=\) "If water is heated to \(100^{\circ} \mathrm{C}\), it turns to vapor." (a) "If Jones told the truth. then if the butler did it, I'll eat my hat." (b) "If the butler did it, then either Jones told the truth or the moon is made of green cheese, but not both." (c) "It is not the case that both Jones told the truth and the moon is made of green cheese." (d) "Jones did not tell the truth, and the moon is not made of green cheese, and I'll not eat my hat." (e) "If Jones told the truth implies I'll eat my hat, then if the butler did it, the moon is made of green cheese." (f) "Jones told the truth, and if water is heated to \(100^{\circ} \mathrm{C}\), it turns to vapor."
Problem 2
The terms of a sequence are given recursively as \(p_{0}=3, p_{1}=7,\) and \(p_{n}=3 p_{n-1}-\) \(2 p_{n-2}\) for \(n \geq 2\). Find the first eight terms of this sequence.
Problem 2
A film class had 33 students who liked Hitchcock movies, 21 students who liked Spielberg movies, and 17 students who liked both kinds of films. How many students were in the class if every student is renresented in the survey?
Problem 2
Find all the clements of (0,1,2,3\\} that, when substituted for \(n,\) satisfy: $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}$$
Problem 2
Translate the following expressions of propositional logic into words using the following translation of the proposition letters: \(p=\) "All the world is apple pie." \(q=\) "All the seas are ink." \(r=\) "All the trees are bread and cheese." \(s=\) There is nothing to drink." \(t=\) "Socrates was a man." \(u=\) "All men are mortal." \(v="\) Socrates was mortal." (a) \((p \wedge q \wedge r) \rightarrow s\) (b) \((t \wedge u) \rightarrow v\) (c) \(\neg s \rightarrow \neg v\) (d) \(p \wedge(q \wedge r) \vee(t \wedge u) \vee(\neg s \vee \neg v)\) \((e)((p \vee t) \wedge(q \vee u)) \leftrightarrow(s \wedge v)\) One must sometimes be a bit creative in using language to make the results comprehensible
Problem 3
The terms of a sequence are given recursively as \(a_{0}=0, a_{1}=4,\) and \(a_{n}=8 a_{n-1}-\) \(16 a_{n-2}\) for \(n \geq 2\). Find the first cight terms of this sequence.
