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Magazine Problem 2
(a) Can a finite, nonempty set be inductive? Explain. (b) Is the empty set inductive? Explain.
Problem 3
For which natural numbers \(n\) is \(n !>3^{n} ?\) Justify your conclusion.
Problem 4
The quadratic formula can be used to show that \(\alpha=\frac{1+\sqrt{5}}{2}\) and \(\beta=\frac{1-\sqrt{5}}{2}\) are the two real number solutions of the quadratic equation \(x^{2}-x-1=0\). Notice that this implies that $$ \begin{array}{l} \alpha^{2}=\alpha+1, \text { and } \\ \beta^{2}=\beta+1 \end{array} $$ It may be surprising to find out that these two irrational numbers are closely related to the Fibonacci numbers. (a) Verify that \(f_{1}=\frac{\alpha^{1}-\beta^{1}}{\alpha-\beta}\) and that \(f_{2}=\frac{\alpha^{2}-\beta^{2}}{\alpha-\beta}\). (b) (This part is optional, but it may help with the induction proof in part (c).) Work with the relation \(f_{3}=f_{2}+f_{1}\) and substitute the expressions for \(f_{1}\) and \(f_{2}\) from part (a). Rewrite the expression as a single fraction and then in the numerator use \(\alpha^{2}+\alpha=\alpha(\alpha+1)\) and a similar equation involving \(\beta .\) Now prove that \(f_{3}=\frac{\alpha^{3}-\beta^{3}}{\alpha-\beta}\).(c) Use induction to prove that for each natural number \(n,\) if \(\alpha=\frac{1+\sqrt{5}}{2}\) and \(\beta=\frac{1-\sqrt{5}}{2},\) then \(f_{n}=\frac{\alpha^{n}-\beta^{n}}{\alpha-\beta} .\) Note: This formula for the \(n^{t h}\) Fibonacci number is known as Binet's formula, named after the French mathematician Jacques Binet ( \(1786-1856\) ).
Problem 4
(a) Verify that \(\left(1-\frac{1}{4}\right)=\frac{3}{4}\) and that \(\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)=\frac{4}{6}\). (b) Verify that \(\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right)=\frac{5}{8}\) and that \(\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right)\left(1-\frac{1}{25}\right)=\frac{6}{10}\) (c) For \(n \in \mathbb{N}\) with \(n \geq 2,\) make a conjecture about a formula for the product \(\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right) \cdots\left(1-\frac{1}{n^{2}}\right) .\) (d) Based on your work in Parts (4a) and (4b), state a proposition and then use the Extended Principle of Mathematical Induction to prove your proposition.
Problem 5
Instead of using induction, we can sometimes use previously proven results about a summation to obtain results about a different summation. (a) Use the result in Progress Check 4.3 to prove the following proposition: For each natural number \(n, 3+6+9+\cdots+3 n=\frac{3 n(n+1)}{2}\) (b) Subtract \(n\) from each side of the equation in Part (a). On the left side of this equation, explain why this can be done by subtracting 1 from each term in the summation. (c) Algebraically simplify the right side of the equation in Part (b) to obtain a formula for the sum \(2+5+8+\cdots+(3 n-1)\). Compare this to Exercise (3a).
Problem 7
In Section \(3.1,\) we defined congruence modulo \(n\) for a natural number \(n,\) and in Section \(3.5,\) we used the Division Algorithm to prove that each integer is congruent, modulo \(n,\) to precisely one of the integers \(0,1,2, \ldots, n-1\) (Corollary 3.32). (a) Find the value of \(r\) so that \(4 \equiv r(\bmod 3)\) and \(r \in\\{0,1,2\\}\). (b) Find the value of \(r\) so that \(4^{2} \equiv r(\bmod 3)\) and \(r \in\\{0,1,2\\}\). (c) Find the value of \(r\) so that \(4^{3} \equiv r(\bmod 3)\) and \(r \in\\{0,1,2\\}\). (d) For two other values of \(n,\) find the value of \(r\) so that \(4^{n} \equiv r(\bmod 3)\) and \(r \in\\{0,1,2\\}\) (e) If \(n \in \mathbb{N},\) make a conjecture concerning the value of \(r\) where \(4^{n} \equiv r(\bmod 3)\) and \(r \in\\{0,1,2\\} .\) This conjecture should be written as a self-contained proposition including an appropriate quantifier. (f) Use mathematical induction to prove your conjecture.
Problem 8
Can each natural number greater than or equal to 4 be written as the sum of at least two natural numbers, each of which is a 2 or a 3 ? Justify your conclusion. For example, \(7=2+2+3,\) and \(17=2+2+2+2+3+3+3\).
Problem 8
Use mathematical induction to prove each of the following: (a) For each natural number \(n, 3\) divides \(\left(4^{n}-1\right)\). (b) For each natural number \(n, 6\) divides \(\left(n^{3}-n\right)\).
Problem 9
Can each natural number greater than or equal to 6 be written as the sum of at least two natural numbers, each of which is a 2 or a 5 ? Justify your conclusion. For example, \(6=2+2+2,9=2+2+5,\) and \(17=2+5+5+5\).
Problem 13
Prove or disprove each of the following propositions: (a) For each \(n \in \mathbb{N}, \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}\). (b) For each natural number \(n\) with \(n \geq 3\), $$ \frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\cdots+\frac{1}{n(n+1)}=\frac{n-2}{3 n+3} $$ (c) For each \(n \in \mathbb{N}, 1 \cdot 2+2 \cdot 3+3 \cdot 4+\cdots+n(n+1)=\frac{n(n+1)(n+2)}{3}\).
