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Suppose that Co, C, C2, ... is a sequence defined as follows: 2, c = 2, c2 = 6, Ck = 30 -3 for every integer k 2 3. Co %3D !3! Prove that c is even for each integer n20.

Suppose that Co, C, C2, ... is a sequence defined as follows: 2, c = 2, c2 = 6, Ck = 30 -3 for every integer k 2 3. Co %3D !3! Prove that c is even for each integer n20.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 55E: The Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is defined recursively by f1=1,f2=1,fn+2=fn+1+fn for...
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Suppose that Co, C, C2, ... is a sequence defined as follows: Co = 2, C = 2, C2 = 6, Ck = 3c,-3 for every integer k 2 3. Prove that co is even for each integer n 20. Proof (by strong mathematical induction): Let the property P(n) be the following sentence. C, is even. We will show that P(n) is true for every integer n 2 0. Show that P(0), P(1), and P(2) are true: V even. P(0), P(1), and P(2) are the statements "Co is even," "c, is even," and "c, is even," respectively. These statements are true because Select- Show that for every integer k > 2, if P(i) is true for each integer i from 0 through k, then P(k + 1) is true: Let k be any integer with k 2 2, and suppose c, is-Select-- v for every integer i with 0sisk. This is the Select- We must show that c+1 is By definition of Co, C1, C2 -- Select- Ck +1= Thus C is Select Since k 2 2, we have that 0sk-2s k. So we can apply the inductive hypothesis to conclude that c is 3 times an Select-
Transcribed Image Text:Suppose that Co, C, C2, ... is a sequence defined as follows: Co = 2, C = 2, C2 = 6, Ck = 3c,-3 for every integer k 2 3. Prove that co is even for each integer n 20. Proof (by strong mathematical induction): Let the property P(n) be the following sentence. C, is even. We will show that P(n) is true for every integer n 2 0. Show that P(0), P(1), and P(2) are true: V even. P(0), P(1), and P(2) are the statements "Co is even," "c, is even," and "c, is even," respectively. These statements are true because Select- Show that for every integer k > 2, if P(i) is true for each integer i from 0 through k, then P(k + 1) is true: Let k be any integer with k 2 2, and suppose c, is-Select-- v for every integer i with 0sisk. This is the Select- We must show that c+1 is By definition of Co, C1, C2 -- Select- Ck +1= Thus C is Select Since k 2 2, we have that 0sk-2s k. So we can apply the inductive hypothesis to conclude that c is 3 times an Select-
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