Ifr is odd, then 2² is odd. If Let z and y be positive numbers. If x ≤y then √√√ and y are positive real numbers then 2√ry ≤a+y N, then 1+ (-1)" (2n-1) is a multiple of 4 I If n the following by counter example: ne Z if n is odd. then n² +n is even.

Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter9: Polynomial And Rational Functions
Section9.2: Remainder And Factor Theorems
Problem 49PS
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1. Use direct proof to prove the following:
(a) If a is odd, then 2² is odd.
(b) Let x and y be positive numbers. If r ≤y then √ √y
(c) If z and y are positive real numbers then 2√xy ≤ x + y
(d) If n € N, then 1+ (-1)" (2n-1) is a multiple of 4
2. Prove the following by counter example:
(a) Vn € Z if n is odd. then n² +n is even.
(b) The difference of any two rational numbers is rational.
(c) The average of any two rational numbers is rational.
Transcribed Image Text:1. Use direct proof to prove the following: (a) If a is odd, then 2² is odd. (b) Let x and y be positive numbers. If r ≤y then √ √y (c) If z and y are positive real numbers then 2√xy ≤ x + y (d) If n € N, then 1+ (-1)" (2n-1) is a multiple of 4 2. Prove the following by counter example: (a) Vn € Z if n is odd. then n² +n is even. (b) The difference of any two rational numbers is rational. (c) The average of any two rational numbers is rational.
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