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Unformatted text preview: Chapter 8: Recursion March 10, 2008 Outline 1 8.1 Recursively Defined Sequences 2 8.2 Solving Recurrence Relations by Iteration 3 8.4 General Recursive Definitions Recursively Defined Sequences • As mentioned in 4.1, a sequence can be defined using recursion . This means that the kth terms is defined using an equation involving previous terms in the sequence. For example, a k = a k 1 + 2 a k 2 , k ≥ 2 and a = 1 , a 1 = 2 • The values for a and a 1 in this example are called the initial values of the sequence. • The formula a k = a k 1 + 2 a k 2 , k ≥ 2 which relates the kth term to some of its predecessors is called the recurrence relation for this sequence. Example For the recursively defined sequence a k = a k 1 + 2 a k 2 , k ≥ 2 with initial conditions a = 1 , a 1 = 2 compute the terms a 2 , a 3 , and a 4 . Solution: a 2 = a 1 + 2 a = 2 + 2 ( 1 ) = a 3 = a 2 + 2 a 1 = + 2 · 2 = 4 a 4 = a 3 + 2 a 2 = 4 + 2 · = 4 Example Show that the sequence , 1 , 5 , 19 , . . . , 3 n 2 n , . . . satisfies the recurrence relation d k = 5 d k 1 6 d k 2 , k ≥ 2 Solution: Substitute d k 1 = 3 k 1 2 k 1 d k 2 = 3 k 2 2 k 2 into the recurrence relation: d k = 5 · ( 3 k 1 2 k 1 ) 6 · ( 3 k 2 2 k 2 ) = 5 · 3 k 1 5 · 2 k 1 6 · 3 k 2 + 6 · 2 k 2 = 5 · 3 k 1 5 · 2 k 1 2 · 3 k 1 + 3 · 2 k 1 = 3 · 3 k 1 2 · 2 k 1 = 3 k 2 k Basic Problem of Recursion Given a recursively defined sequence a n , find an analytical formula a n = f ( n ) for the sequence. In general, this may be a difficult problem. In this course, we will look at relatively simple instances of recursively defined sequences. Our approach will generally consist of the following two steps: 1 Guess an explicit formula for a n = f ( n ) . [Often, this consists of writing out the first several terms and trying to discover a pattern.] 2 Prove that this formula is correct using induction. Example: The Tower of Hanoi • In 1883., Edouard Lucas invented a mathematical puzzle, which he called The Tower of Hanoi. • The setup of the puzzle consisted of 8 disks of different sizes placed in decreasing size on top of one another on one pole in a row of three poles. The task is to transfer these disks from the pole where they lie originally to one of the others, never placing a larger disk on top of a smaller one. Figure: The Tower of Hanoi • In 1884., De Parville described it in a more colourful way, after slightly changing the statement of the original puzzle: “ On the steps of the altar in the temple of Benares, for many, many years Brahmins have been moving a tower of sixtyfour golden disks from one pole to another, one by one, never placing a larger on top of a smaller. When all disks have been transferred, the Tower and the Brahmins will fall, and it will be the end of the world. ” • Assuming this legend is true, is there anything to worry about? Suppose that the poles are labelled as A , B , and C . We will approach this problem in a general way and assume that, at the...
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 Spring '08
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 Recursion, Recurrence relation, recursively defined sequence

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