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math1005ass2sol

math1005ass2sol - 40 marks MATH1005A Summer2009 Assignment...

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40 marks MATH1005A, Summer2009 Assignment 2 (solution) Problem 1 (10 marks) . Show that the sequence defined by a 1 = 2 , a n +1 = 1 3 - a n satisfies 0 < a n 2 and is decreasing. Deduce that the the sequence is convergent and find its limit. Solution: We start by proving that a n 2 by mathematical induction. a 1 = 2 2 . Assume a n 2 we prove that a n +1 2. a n 2 3 - a n 3 - 2 = 1 a n +1 = 1 3 - a n 1 < 2 . So by the principle of induction a n 2 for all n . Since a n 2, we have 3 - a n > 0. Therefore a n +1 = 1 / (3 - a n ) 0 for all n . Moreover a 1 = 2 > 0. So 0 < a n 2 fo all n . To prove that { a n } is decreasing we use induction again. a 1 = 2 > 1 = 1 3 - 2 = a 2 . Assume a n - 1 > a n we prove that a n > a n +1 . a n - 1 > a n 3 - a n - 1 < 3 - a n a n = 1 3 - a n - 1 > 1 3 - a n = a n +1 . So by the principle of induction a n > a n +1 for all n . Now by Monotonic Sequence Theorem the sequence { a n } is convergent. Assume that lim n →∞ a n = L , Where L is a real number. Then lim n →∞ a n +1 = L . Thus L = lim n →∞ a n +1 = lim n →∞ 1 3 - a n = 1 3 - lim n

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