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Suppose that Co C Cg... is a sequence defined as follows: o = 2, c, = 2, c, = 6, C = 30,- for every integer k 2 3. Prove that c, is even for each integer n2 0. Proof (by strong mathematical induction): Let the property P(n) be the following sentence. 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: 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-- even. Show that for every integer k 2 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 0s isk. This is the --Select- We must show that +1 is --Select--- . By definition of Co C. C2.. Ck +1 Since k z 2, we have that 0 sk-2s k. So we can apply the inductive hypothesis to conclude that c1 is 3 times an -Select---v Thus Ck +1 -Select-- is

Suppose that Co C Cg... is a sequence defined as follows: o = 2, c, = 2, c, = 6, C = 30,- for every integer k 2 3. Prove that c, is even for each integer n2 0. Proof (by strong mathematical induction): Let the property P(n) be the following sentence. 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: 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-- even. Show that for every integer k 2 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 0s isk. This is the --Select- We must show that +1 is --Select--- . By definition of Co C. C2.. Ck +1 Since k z 2, we have that 0 sk-2s k. So we can apply the inductive hypothesis to conclude that c1 is 3 times an -Select---v Thus Ck +1 -Select-- is

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 Cơ C C2.. is a sequence defined as follows: o = 2, c, = 2, c, = 6, = 30 -3 for every integer k 2 3. Prove that c, is even for each integer n 2 0. Proof (by strong mathematical induction): Let the property P(n) be the following sentence. C, 5 even. is We will show that P(n) is true for every integer n2 0. Show that P(0), P(1), and P(2) are true: P(0), P(1), and P(2) are the statements "c, is even," "c, is even," and "c, is even," respectively. These statements are true because -Select- even. Show that for every integer k2 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 is |--Select--- v for every integer i with 0sisk. This is the ---Select-- We must show that c,, is -Select- V. By definition of co C Czr -... CK +1= Since k 2 2, we have that 0s k - 2 s k. So we can apply the inductive hypothesis to conclude that c41 is 3 times an -Select-- v. Thus c+ is select--- V
Transcribed Image Text:Suppose that Cơ C C2.. is a sequence defined as follows: o = 2, c, = 2, c, = 6, = 30 -3 for every integer k 2 3. Prove that c, is even for each integer n 2 0. Proof (by strong mathematical induction): Let the property P(n) be the following sentence. C, 5 even. is We will show that P(n) is true for every integer n2 0. Show that P(0), P(1), and P(2) are true: P(0), P(1), and P(2) are the statements "c, is even," "c, is even," and "c, is even," respectively. These statements are true because -Select- even. Show that for every integer k2 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 is |--Select--- v for every integer i with 0sisk. This is the ---Select-- We must show that c,, is -Select- V. By definition of co C Czr -... CK +1= Since k 2 2, we have that 0s k - 2 s k. So we can apply the inductive hypothesis to conclude that c41 is 3 times an -Select-- v. Thus c+ is select--- V
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