2.4: Exercises
Prove that
\[ 1^2 + 2^2 + \cdots + n^2 = \frac{n(n + 1)(2n + 1)}{6} \nonumber \]
for \(n \in {\mathbb N}\text{.}\)
Prove that
\[ 1^3 + 2^3 + \cdots + n^3 = \frac{n^2(n + 1)^2}{4} \nonumber \]
for \(n \in {\mathbb N}\text{.}\)
Prove that \(n! \gt 2^n\) for \(n \geq 4\text{.}\)
Prove that
\[ x + 4x + 7x + \cdots + (3n - 2)x = \frac{n(3n - 1)x}{2} \nonumber \]
for \(n \in {\mathbb N}\text{.}\)
Prove that \(10^{n + 1} + 10^n + 1\) is divisible by \(3\) for \(n \in {\mathbb N}\text{.}\)
Prove that \(4 \cdot 10^{2n} + 9 \cdot 10^{2n - 1} + 5\) is divisible by \(99\) for \(n \in {\mathbb N}\text{.}\)
Show that
\[ \sqrt[n]{a_1 a_2 \cdots a_n} \leq \frac{1}{n} \sum_{k = 1}^{n} a_k\text{.} \nonumber \]
Prove the Leibniz rule for \(f^{(n)} (x)\text{,}\) where \(f^{(n)}\) is the \(n\)th derivative of \(f\text{;}\) that is, show that
\[ (fg)^{(n)}(x) = \sum_{k = 0}^{n} \binom{n}{k} f^{(k)}(x) g^{(n - k)}(x)\text{.} \nonumber \]
Use induction to prove that \(1 + 2 + 2^2 + \cdots + 2^n = 2^{n + 1} - 1\) for \(n \in {\mathbb N}\text{.}\)
Prove that
\[ \frac{1}{2}+ \frac{1}{6} + \cdots + \frac{1}{n(n + 1)} = \frac{n}{n + 1} \nonumber \]
for \(n \in {\mathbb N}\text{.}\)
If \(x\) is a nonnegative real number, then show that \((1 + x)^n - 1 \geq nx\) for \(n = 0, 1, 2, \ldots\text{.}\)
Power Sets
Let \(X\) be a set. Define the power set of \(X\text{,}\) denoted \({\mathcal P}(X)\text{,}\) to be the set of all subsets of \(X\text{.}\) For example,
\[ {\mathcal P}( \{a, b\} ) = \{ \emptyset, \{a\}, \{b\}, \{a, b\} \}\text{.} \nonumber \]
For every positive integer \(n\text{,}\) show that a set with exactly \(n\) elements has a power set with exactly \(2^n\) elements.
Prove that the two principles of mathematical induction stated in Section 2.1 are equivalent.
Show that the Principle of Well-Ordering for the natural numbers implies that 1 is the smallest natural number. Use this result to show that the Principle of Well-Ordering implies the Principle of Mathematical Induction; that is, show that if \(S \subset {\mathbb N}\) such that \(1 \in S\) and \(n + 1 \in S\) whenever \(n \in S\text{,}\) then \(S = {\mathbb N}\text{.}\)
For each of the following pairs of numbers \(a\) and \(b\text{,}\) calculate \(\gcd(a,b)\) and find integers \(r\) and \(s\) such that \(\gcd(a,b) = ra + sb\text{.}\)
- \(14\) and \(39\)
- \(234\) and \(165\)
- \(1739\) and \(9923\)
- \(471\) and \(562\)
- \(23771\) and \(19945\)
- \(-4357\) and \(3754\)
Let \(a\) and \(b\) be nonzero integers. If there exist integers \(r\) and \(s\) such that \(ar + bs =1\text{,}\) show that \(a\) and \(b\) are relatively prime.
Fibonacci Numbers
The Fibonacci numbers are
\[ 1, 1, 2, 3, 5, 8, 13, 21, \ldots\text{.} \nonumber \]
We can define them inductively by \(f_1 = 1\text{,}\) \(f_2 = 1\text{,}\) and \(f_{n + 2} = f_{n + 1} + f_n\) for \(n \in {\mathbb N}\text{.}\)
- Prove that \(f_n \lt 2^n\text{.}\)
- Prove that \(f_{n + 1} f_{n - 1} = f^2_n + (-1)^n\text{,}\) \(n \geq 2\text{.}\)
- Prove that \(f_n = [(1 + \sqrt{5}\, )^n - (1 - \sqrt{5}\, )^n]/ 2^n \sqrt{5}\text{.}\)
- Show that \(\phi = \lim_{n \rightarrow \infty} f_{n + 1} / f_n = (\sqrt{5} + 1)/2\text{.}\) The constant \(\phi\) is known as the golden ratio .
- Prove that \(f_n\) and \(f_{n + 1}\) are relatively prime.
Let \(a\) and \(b\) be integers such that \(\gcd(a,b) = 1\text{.}\) Let \(r\) and \(s\) be integers such that \(ar + bs = 1\text{.}\) Prove that
\[ \gcd(a,s) = \gcd(r,b) = \gcd(r,s) = 1\text{.} \nonumber \]
Let \(x, y \in {\mathbb N}\) be relatively prime. If \(xy\) is a perfect square, prove that \(x\) and \(y\) must both be perfect squares.
Using the division algorithm, show that every perfect square is of the form \(4k\) or \(4k + 1\) for some nonnegative integer \(k\text{.}\)
Suppose that \(a, b, r, s\) are pairwise relatively prime and that
\begin{align*} a^2 + b^2 & = r^2\\ a^2 - b^2 & = s^2\text{.} \end{align*}
Prove that \(a\text{,}\) \(r\text{,}\) and \(s\) are odd and \(b\) is even.
Let \(n \in {\mathbb N}\text{.}\) Use the division algorithm to prove that every integer is congruent mod \(n\) to precisely one of the integers \(0, 1, \ldots, n-1\text{.}\) Conclude that if \(r\) is an integer, then there is exactly one \(s\) in \({\mathbb Z}\) such that \(0 \leq s \lt n\) and \([r] = [s]\text{.}\) Hence, the integers are indeed partitioned by congruence mod \(n\text{.}\)
Define the least common multiple of two nonzero integers \(a\) and \(b\text{,}\) denoted by \(\operatorname{lcm}(a, b)\text{,}\) to be the nonnegative integer \(m\) such that both \(a\) and \(b\) divide \(m\text{,}\) and if \(a\) and \(b\) divide any other integer \(n\text{,}\) then \(m\) also divides \(n\text{.}\) Prove there exists a unique least common multiple for any two integers \(a\) and \(b\text{.}\)
If \(d= \gcd(a, b)\) and \(m = \operatorname{lcm}(a, b)\text{,}\) prove that \(dm = |ab|\text{.}\)
Show that \(\operatorname{lcm}(a, b) = ab\) if and only if \(\gcd(a,b) = 1\text{.}\)
Prove that \(\gcd(a,c) = \gcd(b,c) =1\) if and only if \(\gcd(ab,c) = 1\) for integers \(a\text{,}\) \(b\text{,}\) and \(c\text{.}\)
Let \(a, b, c \in {\mathbb Z}\text{.}\) Prove that if \(\gcd(a,b) = 1\) and \(a \mid bc\text{,}\) then \(a \mid c\text{.}\)
Let \(p \geq 2\text{.}\) Prove that if \(2^p - 1\) is prime, then \(p\) must also be prime.
Prove that there are an infinite number of primes of the form \(6n + 5\text{.}\)
Prove that there are an infinite number of primes of the form \(4n - 1\text{.}\)
Using the fact that \(2\) is prime, show that there do not exist integers \(p\) and \(q\) such that \(p^2 = 2 q^2\text{.}\) Demonstrate that therefore \(\sqrt{2}\) cannot be a rational number.