91Ó°ÊÓ

Prove each statement in 8-23 by mathematical induction. \(1+3 n \leq 4^{n}\), for every integer \(n \geq 0\).

Use the well-ordering principle to prove Theorem \(3.3 .2\); Every integer greater than 1 is divisible by a prime number.

Compute the summations and products in 19-28 $$ \sum_{k=1}^{5}(k+1) $$

$$ 5+10+15+20+\cdots+300 $$

The Archimedean property for the rational numbers states that for all rational numbers \(r\), there is an integer \(n\) such that \(n>r\). Prove this property.

$$ 3+4+5+6+\cdots+1000 $$

$$ 7+8+9+10+\cdots+600 $$

Use the well-ordering principle to prove that given any integer \(n \geq 1\), there exists an odd integer \(m\) and a nonnegative integer \(k\) such that \(n=2^{k} \cdot m\).

Use the well-ordering principle to prove that if \(a\) and \(b\) are any integers not both zero, then there exist integers \(u\) and \(v\) such that \(\operatorname{gcd}(a, b)=u a+v b\). (Hint: Let \(S\) be the set of all positive integers of the form \(u a+v b\) for some integers \(u\) and \(v .)\)

Prove that for all integers \(n \geq 1\), $$ \begin{aligned} \frac{1}{3} &=\frac{1+3}{5+7}=\frac{1+3+5}{7+9+11}=\cdots \\ &=\frac{1+3+\cdots+(2 n-1)}{(2 n+1)+\cdots+(4 n-1)} \end{aligned} $$