Unformatted text preview: a n +1 and b n +1 are obtained from the terms a n and b n , and use this rule to ﬁnd the terms a 7 , b 7 , a 8 and b 8 . (ii) Use a calculator to ﬁnd the ratio b n 2 /a n 2 for 1 ≤ n ≤ 6 and make a conjecture about the limit lim n →∞ ( b n /a n ). (b) For a real number x , let b x c denote the largest integer which is less than or equal to x . For each positive integer n , let a n = ± n √ 2 ² and let b n be the n th positive integer which does not occur in the sequence { a n } . The ﬁrst 20 terms of the sequence { a n } and the ﬁrst 8 terms of the sequence { b n } are listed below. n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a n 1 2 4 5 7 8 9 11 12 14 15 16 18 19 21 22 24 25 26 28 b n 3 6 10 13 17 20 23 27 Make a conjecture about a rule which allows us to determine the terms a n and b n without using a calculator and without using the value of √ 2....
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 Spring '08
 ANDREWCHILDS
 Algebra, Real Numbers, Integers, Prime number, odd prime numbers

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