Homework1.pdf - Seul-Ki Kim ICS 6D Homework 1 1 Exercise...

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Seul-Ki Kim ICS 6D January 16, 2019 Homework 1 1. Exercise 8.1.1 : Evaluating sequences. Give the first ten terms of the following sequences. You can assume that the sequences start with an index of 1. Logs are to base 2. Indicate whether the sequence is increasing, decreasing, non-increasing, or non- decreasing. The sequence may have more than one of those properties. ------------------------------------------------------------------------------------------------------------ (a) The n th tern is n . 1 , 2 ⌉, ⌈ 3 ⌉, ⌈ 4 ⌉, ⌈ 5 ⌉, ⌈ 6 ⌉, ⌈ 7 ⌉, ⌈ 8 ⌉, ⌈ 9 ⌉¸⌈ 10 ⌈1⌉ , ⌈1.4⌉, ⌈1.7⌉, ⌈2⌉, ⌈2.2⌉, ⌈ 2 .4⌉, ⌈ 2 .6⌉, ⌈ 2 .8⌉, ⌈3⌉¸⌈ 3 .2⌉ 1, 2, 2, 2, 3, 3, 3, 3, 3, 4 Non-decreasing but not increasing (b) The first two terms in the sequence are 1. The rest of the terms are the sum of the two preceding terms. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 Non-decreasing but not increasing (c) The n th term is the largest integer k such that k! ≤ n. 1, 2, 2, 2, 2, 3, 3, 3, 3, 3 For example, the 4 th term 2! = 2 ≤ 4 3! = 6 > 4 So for the 4 th term, we will take 2 Non-decreasing but not increasing (d) The n th term is 1/n. 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10 Decreasing and non-increasing (e) The n th term is 3. 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 Non-increasing and non-decreasing (f) The n th term is n 2 . 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 Increasing and non-decreasing (g) The n th term is logn . (base 2) log1 ⌉, ⌈ log2 ⌉, ⌈ log3 ⌉, ⌈ log4 ⌉, ⌈ log5 ⌉¸⌈ log6 ⌉, ⌈ log7 ⌉, ⌈ log8 ⌉¸ ⌈ log9 ⌉, ⌈ log10 0 ⌉, ⌈ 1 ⌉, ⌈ 1.6 ⌉, ⌈ 2 ⌉, ⌈ 2.3 ⌉¸⌈ 2.6 ⌉, ⌈ 2.8 ⌉, ⌈ 3 ⌉¸ ⌈ 3.2 ⌉, ⌈ 3.3 0, 1, 2, 2, 3, 3, 3, 3, 4, 4 Non-decreasing but not increasing (h) The n th term is 2 logn . (base 2) 2 log1 , 2 log2 , 2 log3 , 2 log4 , 2 log5 , 2 log6 , 2 log7 , 2 log8 , 2 log9 , 2 log10 2 0 , 2 1 , 2 2 , 2 2 , 2 3 , 2 3 , 2 3 , 2 3 , 2 4 , 2 4 1, 2, 4, 4, 8, 8, 8, 8, 16, 16 Non-decreasing but not increasing
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(i) The n th term is ି௡ .
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  • Fall '07
  • Jarecki
  • Summation, Geometric progression

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