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Show that Pn satisfy the following recursion Pn+1 = P&Pn-k- k=0

Show that Pn satisfy the following recursion Pn+1 = P&Pn-k- k=0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let us consider parenthesis, as used in a sentence for the written English language. The only rule for a single parenthesis is that it must open, we write " (" for that, and it must close, in which case we write ")". If we use a parenthesis inside another parenthesis, we must make sure that we close the parenthesis inside first, before closing the external parenthesis. Otherwise it is not correctly written. For instance, 0) is correct, but )(() is not correct. Let Pn be the number of ways in which n parenthesis can be written correctly. The start of the sequence is (Ро, Р, Ра, Рз, ...) %3D (1, 1, 2, 5,...), corresponding to no parenthesis, which gives Po = 1, the unique parenthesis (), which gives P = 1, P2 = 2 because we can write ()) and (()) and finally there are five ways, P3 = 5, of correctly writing MAT 108: PROBLEM SET 3 3 three parenthesis: )00,0(0),(0)0), (O0),((O)). Show that Pn satisfy the following recursion n Pn+1 => Pk Pn-k- k=0
Transcribed Image Text:Let us consider parenthesis, as used in a sentence for the written English language. The only rule for a single parenthesis is that it must open, we write " (" for that, and it must close, in which case we write ")". If we use a parenthesis inside another parenthesis, we must make sure that we close the parenthesis inside first, before closing the external parenthesis. Otherwise it is not correctly written. For instance, 0) is correct, but )(() is not correct. Let Pn be the number of ways in which n parenthesis can be written correctly. The start of the sequence is (Ро, Р, Ра, Рз, ...) %3D (1, 1, 2, 5,...), corresponding to no parenthesis, which gives Po = 1, the unique parenthesis (), which gives P = 1, P2 = 2 because we can write ()) and (()) and finally there are five ways, P3 = 5, of correctly writing MAT 108: PROBLEM SET 3 3 three parenthesis: )00,0(0),(0)0), (O0),((O)). Show that Pn satisfy the following recursion n Pn+1 => Pk Pn-k- k=0
Expert Solution
Step 1

To show : Pn satisfy the recursion

∑k=0nPkPn-k=Pn+1   ...i

it is given in the problem that

P0=1 , P1=1 , P2=2 , P3=5

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