X7. We know that given any real numbers x1 and x2, if x1X2 0, then x1 0 or x2 = 0. Since there are two factors, let's call this statement P(2). We assume P(2) is true without proving it. a. Use an inductive idea to carefully explain why given any real numbers x1, X2, and x3, if X1X2X3 = 0, then x1 0, x2 0, or x3 0. Be sure to use only P(2) in your argument. Hint: x1x2x3 = (x,x2)x3, which consists of two factors, x1x2, and x3. b. Use an inductive idea to carefully explain why given any real numbers x1, X2, X3, and x4, if x1x2X3X4 = 0, then x1 = 0, x2 = 0, x3 0, or x4 0. Be sure to use only P(2) or P(3) in your argument. c. Let P(n) be the statement that given any n real numbers x1, x2, .,Xn, if x1X2 Xn = 0, then x 0 for some i. Use mathematical induction to prove P(n) is true for all n 2 2. The base case does not need to be proven, but at least acknowledge its truth.
X7. We know that given any real numbers x1 and x2, if x1X2 0, then x1 0 or x2 = 0. Since there are two factors, let's call this statement P(2). We assume P(2) is true without proving it. a. Use an inductive idea to carefully explain why given any real numbers x1, X2, and x3, if X1X2X3 = 0, then x1 0, x2 0, or x3 0. Be sure to use only P(2) in your argument. Hint: x1x2x3 = (x,x2)x3, which consists of two factors, x1x2, and x3. b. Use an inductive idea to carefully explain why given any real numbers x1, X2, X3, and x4, if x1x2X3X4 = 0, then x1 = 0, x2 = 0, x3 0, or x4 0. Be sure to use only P(2) or P(3) in your argument. c. Let P(n) be the statement that given any n real numbers x1, x2, .,Xn, if x1X2 Xn = 0, then x 0 for some i. Use mathematical induction to prove P(n) is true for all n 2 2. The base case does not need to be proven, but at least acknowledge its truth.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 35E
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