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Unformatted text preview: (10%) (ii) Prove that there is some number x such that x 5 + 40 x 2 + cos 2 x + 1 = 10. (12%) 5. Recall the Fibonacci Sequence { F n } is dened recursively by F 1 = 1, F 2 = 1, and F n + 2 = F n + 1 + F n for all n 1. We dene the Lucas Sequence { L n } recursively as follows: L 1 = 1 , L 2 = 3 , L n + 2 = L n + 1 + L n for all n 1 . Prove for all positive integers n 2 that L n = F n1 + F n + 1 . 1 6. (5%) (a) State the Least Upper Bound Axiom. (5%) (b) Suppose S is a set of real numbers that obeys the following properties: (P1) S 6 = R . (P2) S 6 = . (P3) If x S and y < x , then y S . Prove that lub S exists. 2...
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 Spring '08
 UPPAL
 Calculus

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