m25-hw5

# m25-hw5 - F 1 = 1, F 2 = 1 deﬁned by the recurrence...

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Math 25: Advanced Calculus Fall 2010 Homework # 5 Due Date: Friday, October 29 Reading: Finish reading section 1.10. Read sections 2.2 and 2.4. Complete the following problems: 5.1: Problem 1.10.2 from the textbook. 5.2: In class, we saw that for all real numbers a and b , | a + b | ≤ | a | + | b | . Give a careful proof of this fact by examining all possible cases ( a is positive or negative and b is positive or negative) that uses only the deﬁnition of the absolute value function and the ordered ﬁeld axioms. 5.3: Problem 2.4.1 from the textbook. 5.4: Problem 2.4.2 from the textbook. 5.5: Prove that lim n →∞ n 2 does not exist. 5.6: Problem 2.4.10 from the textbook. 5.7: Consider the sequence F 0 = 0,
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Unformatted text preview: F 1 = 1, F 2 = 1 deﬁned by the recurrence relation F n = F n-1 + F n-2 for all n ≥ 2. (a). Write out the ﬁrst 10 terms of this sequence. (b). Use mathematical induction to show that F n = 1 √ 5 " 1 + √ 5 2 ! n-1-√ 5 2 ! n # . In the base case of this problem, you need to establish that the desired formula holds for n = 0 and n = 1; for the inductive step, you need to assume that F n-1 and F n-2 can be expressed according to this formula and then conclude that F n can also be expressed according to this formula. [Hint: Compute ± 1+ √ 5 2 ² 2 and ± 1-√ 5 2 ² 2 before you start] 1...
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## This note was uploaded on 03/18/2012 for the course MAT MAT 25 taught by Professor Stevenklee during the Fall '10 term at UC Davis.

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