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2. If i3n+1 = -1, where i = √√-1, then which of the following is a solution for n? (1) n = 1 (3) n = 3 (2) n = 2 (4) n = 4

2. If i3n+1 = -1, where i = √√-1, then which of the following is a solution for n? (1) n = 1 (3) n = 3 (2) n = 2 (4) n = 4

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Question 2: Complex Numbers** Given \( i^{3n+1} = -1 \), where \( i = \sqrt{-1} \), determine which of the following values of \( n \) is a solution. ### Possible Answers: 1. \( n = 1 \) 2. \( n = 2 \) 3. \( n = 3 \) 4. \( n = 4 \) ### Explanation: To solve this, recall the powers of \( i \): - \( i^1 = i \) - \( i^2 = -1 \) - \( i^3 = -i \) - \( i^4 = 1 \) - \( i^5 = i \) and so forth, repeating every four terms. Use this repetition to test each given \( n \) value. To confirm: 1. For \( n = 1 \): \( i^{3(1)+1} = i^4 = 1 \) (which is not -1) 2. For \( n = 2 \): \( i^{3(2)+1} = i^7 = i^3 = -i \) (which is not -1) 3. For \( n = 3 \): \( i^{3(3)+1} = i^{10} = i^2 = -1 \), which is indeed correct. 4. For \( n = 4 \): \( i^{3(4)+1} = i^{13} = i \) (which is not -1) Thus, the correct value is: \[ \boxed{n = 3} \] This exercise assesses understanding of complex number powers and cyclicity.
Transcribed Image Text:**Question 2: Complex Numbers** Given \( i^{3n+1} = -1 \), where \( i = \sqrt{-1} \), determine which of the following values of \( n \) is a solution. ### Possible Answers: 1. \( n = 1 \) 2. \( n = 2 \) 3. \( n = 3 \) 4. \( n = 4 \) ### Explanation: To solve this, recall the powers of \( i \): - \( i^1 = i \) - \( i^2 = -1 \) - \( i^3 = -i \) - \( i^4 = 1 \) - \( i^5 = i \) and so forth, repeating every four terms. Use this repetition to test each given \( n \) value. To confirm: 1. For \( n = 1 \): \( i^{3(1)+1} = i^4 = 1 \) (which is not -1) 2. For \( n = 2 \): \( i^{3(2)+1} = i^7 = i^3 = -i \) (which is not -1) 3. For \( n = 3 \): \( i^{3(3)+1} = i^{10} = i^2 = -1 \), which is indeed correct. 4. For \( n = 4 \): \( i^{3(4)+1} = i^{13} = i \) (which is not -1) Thus, the correct value is: \[ \boxed{n = 3} \] This exercise assesses understanding of complex number powers and cyclicity.
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