hw1-solution1 - AMATH 383 Introduction to Continuous...

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AMATH 383 Introduction to Continuous Mathematical Modeling Winter, 2012 AMATH 383 Homework Assignment #1: Solution 1. The first 4 numbers in Fibonacci sequence F n with n = 1 , 2 , · · · are 1 , 2 , 3 , 5 . Find the values for a, b, c, d in G n = an + bn 2 + cn 3 + dn 4 , such that the first four G n fit the first four F n . What is F 5 ? and what is G 5 ? Compare F n and G n for 0 n 100 in a graph. Solution: Solving a, b, c, d from G (1) = a + b + c + d = 1 , G (2) = 2 a + 4 b + 8 c + 16 d = 2 , G (3) = 3 a + 9 b + 27 c + 81 d = 3 , G (4) = 4 a + 16 b + 64 c + 256 d = 5 . By Gaussian elimination: a + b + c + d = 1 , 6 b + 24 c + 78 d = 0 , 6 c + 36 d = 0 , 24 d = 1 . Therefore, a = 3 4 , b = 11 24 , c = - 1 4 , d = 1 24 . F 5 = 8 and G 5 = 10 . See Figure 1. 2. Exercise 2 of Chapter 1. Solution: (a) Golden ratio Φ = 1 2 ( 1 + 5 ) 1 . 6180339887 · · · . It is an irrational number satis- fies the equation x 2 = x + 1 . Re-arrange this equation to x = 1 + 1 /x . This implies Φ satisfies Φ = 1 + 1 Φ = 1 + 1 1 + 1 Φ = 1 + 1 1 + 1 1+ 1 Φ = 1 + 1 1 + 1 1+ 1 1+ 1 Φ = 1 + 1 1 + 1 1+ 1 1+ 1 1+ 1 1+ ··· Prof. Hong Qian January 5, 2012
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AMATH 383 Introduction to Continuous Mathematical Modeling Winter, 2012 1E+10 1E+15 1E+20 F(n) G(n) 1 100000 0 25 50 75 100 Figure 1: in which the · · · is Φ - 1 = 1 Φ . (b) Similar to (a), 2 = x - 1 in which x satisfies equation x = 2 + 1 /x . Hence, x = 2 + 1 x = 2 + 1 2 + 1 x = 2 + 1 2 + 1 2+ 1 x = 2 + 1 2 + 1 2+ 1 2+ 1 x = 2 + 1 2 + 1 2+ 1 2+ 1 2+ ··· in which · · · = 1 /x . Therefore, 2 = 1 + 1 x = 1 + 1 2 + 1 x = 1 + 1 2 + 1 2+ 1 x = 1 + 1 2 + 1 2+ 1
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  • Winter '07
  • JUZWIAK,WILLIAM
  • Characteristic polynomial, Continuous Mathematical Modeling, Prof. Hong Qian

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