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# Solved: List the first 10 terms of each of these

ISBN: 9780073383095 37

## Solution for problem 6E Chapter 2.4

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition

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Problem 6E

List the first 10 terms of each of these sequences.a) the sequence obtained by starting with 10 and obtaining each term by subtracting 3 from the previous term________________b) the sequence whose nth term is the sum of the first n positive integers________________c) the sequence whose nth term is ________________d) the sequence whose nth term is ________________e) the sequence whose first two terms are 1 and 5 and each succeeding term is the sum of the two previous terms________________f) the sequence whose nth term is the largest integer whose binary expansion (defined in Section 4.2) has n bits (Write your answer in decimal notation.)________________g) the sequence whose terms are constructed sequentially as follows: start with 1, then add 1, then multiply by 1, then add 2, then multiply by 2, and so on________________h) the sequence whose nth term is the largest integer k such that k! ? n

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SolutionStep-1: In this problem we need to find the first 10 terms of the sequences.a)Given first term of the sequence is ten. That is , . Consider , the term of the sequence , since each term by subtracting 3 from the previous term . If n = 2, then . If n = 3, then . If n = 4, then . If n = 5, then . If n = 6, then . If n = 7, then . If n = 8, then . If n = 9, then . If n = 10, then . Therefore , the first ten terms of the sequence is 10 , 7,4, 1, -2 , -5 , -8 , -11, -14 and -17.Step-2:b)term = sum of the first n positive integers. = 1+2+3+.....+n That is , .If n =1 , then .If n =2 , then .If n =3 , then .If n =4 , then .If n =5 , then .If n =6 , then .If n =7 , then .If n =8 , then .If n = 9, then .If n =10 , then .Therefore , the first 10 terms of the sequence is : 1 , 3, 6, 10 , 15 , 21, 28 , 36 , 45 and 55.Step-3:c) term of the sequence .If n = 1, then .If n = 2, then .If n = 3, then .If n = 4, then .If n = 5, then .If n = 6, then .If n = 7, then .If n = 8, then .If n = 9, then .If n = 10, then .Therefore , the first 10 terms of the sequence is : 1 , 5, 19 , 65 , 211, 665 , 2059 , 6305 , 19171 and 58025.Step-4:d)term is : is a floor function.Note: Therefore , the first 10 terms of the sequence is : 1, 1,1,2,2,2,2,2 ,3 and 3.Step-5: e) The first two terms of the sequence is 1 and 5. That is , . , since each succeeding term is the sum of the two preceding terms. . . . .Therefore , the 10 terms of the sequence is : 1 , 5, 6,11,17,28,45,73,118, and 191.Step-6:f) term = Largest integer whose binary expansion has n bits. We know that , , where n is a non positive integer , . If n = 1, then If n = 2, then If n = 3, then If n = 4, then . If n = 5, then If n = 6, then = 0+2+4+8+16+32+1 = 63. If n = 7, then = 0+2+4+8+16+32+64+1 = 127. If n = 8, then = 0+2+4+8+16+32+64+128+1 = 255. If n = 9, then = 0+2+4+8+16+32+64+128+256+1 = 511. If n = 10, then = = =0+2+4+8+16+32+64+128+256+512+1 = 1023.Therefore , the first 10 terms of the sequence is :1 , 3, 7, 15 , 31, 63 , 127 , 255 , 511 and 1023.Step-7:g) First term of the sequence is one . That is , The second term , since given.Third term , since given .Fourth term Fifth term Sixth term Seventh term Eighth term Ninth term Tenth term Therefore , the first ten terms of the sequence is : 1 , 2, 2, 4, 8, 11, 33, 37 , 148 and 153.Step-8:h)term = largest integer k such that , since , since , since , since , since , since , since , since , since Therefore , the first 10 terms of the sequence is : 1 , 2, 2, 2, 2, 3, 3, 3,3 and 3.

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Solved: List the first 10 terms of each of these