8. Determine all factors of 4 - 1.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.4: Mathematical Induction
Problem 25E
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Section 1: Numbers
1. Let za = 2 and define
1+21n
(a) In+1 =
for n20. Prove that +1 < In, for all n.
1+.
2+ In for n > 0. Prove that zan #2{m=1): Tan+1 2 Im-1; for n 2 1.
1+ In
(b) 2n+1 =
2. Does there exist n e Z such that ged(2n + 3, 5n + 7) + 1?
3. Prove that the system 15z + 12y = 6 has a solution for r, y €Z if and only if 3 divides b.
Transcribed Image Text:Section 1: Numbers 1. Let za = 2 and define 1+21n (a) In+1 = for n20. Prove that +1 < In, for all n. 1+. 2+ In for n > 0. Prove that zan #2{m=1): Tan+1 2 Im-1; for n 2 1. 1+ In (b) 2n+1 = 2. Does there exist n e Z such that ged(2n + 3, 5n + 7) + 1? 3. Prove that the system 15z + 12y = 6 has a solution for r, y €Z if and only if 3 divides b.
4. If the roots of the equation ar + bz +e = 0, a # 0 differ by 2 then prove that 6 = 4sa(a +e).
5. If one root of the equation ar + br +e = 0, a ± 0 is equal to twice the other root, prove that
6 - 9ae.
6. Let p be a prime number. Factorize a - 1.
7. Find all factors of a - 1.
8. Determine all factors of a4 - 1.
Transcribed Image Text:4. If the roots of the equation ar + bz +e = 0, a # 0 differ by 2 then prove that 6 = 4sa(a +e). 5. If one root of the equation ar + br +e = 0, a ± 0 is equal to twice the other root, prove that 6 - 9ae. 6. Let p be a prime number. Factorize a - 1. 7. Find all factors of a - 1. 8. Determine all factors of a4 - 1.
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