Which of the following statements about rational numbers are true? [More than one ofthe choices may be true.]p+Vp, qe Q, EQiVp € Qt, ne Zt, Σ=1 ;n(n+1)РFor every non-empty set S of positive rational numbers, there exists a smallest element in theset S.\p € Qt,p #1,neZ,Σ on =²+1p-11

Answered: Which of the following statements about…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

Which of the following statements about rational numbers are true? [More than one of
the choices may be true.]
p+
Vp, qe Q, EQ
i
Vp € Qt, ne Zt, Σ=1 ;
n(n+1)
Р
For every non-empty set S of positive rational numbers, there exists a smallest element in the
set S.
\p € Qt,p #1,neZ,Σ on =
²+1
p-1
1

Expert Solution

Step 1

Introduction:

The set of rational numbers is a subset of the set of real numbers. In the most general way, a rational number is formulated by diving two integers. The denominator must be a non-zero integer.

Given:

The given statements are:

$(a) . \forall p, q \in ℚ, \frac{p + q}{2} \in ℚ (b) . \forall p \in ℚ^{+}, n \in ℤ^{+}, \sum_{i = 1}^{n} \frac{i}{p} = \frac{n (n + 1)}{p} (c) . For every non-empty set S of positive rational numbers, there exists a smallest element in the set S . (d) . \forall p \in ℚ^{+}, p \neq 1, n \in ℤ^{\geq 0}, \sum_{i = 0}^{n} p^{i} = \frac{p^{n + 1} - 1}{p - 1}$