Number Puzzles and Sequences

Transcription

1 Number Puzzles and Sequences Reading: EC 1.2 Peter J. Haas INFO 150 Fall Semester 2018 Lecture 2 1/ 16

2 Overview and Examples Guess the Next Number Sequences and Sequence Notation Discovering Patterns in Sequences Sums Lecture 2 2/ 16

3 Guess the Next Number ?? ?? ?? ?? ?? Why do we care? I Training for recursive thinking in a simple setting I Used later when learning how to write proofs I Diagnosing time and space complexity of computations I At each time step each process spawns two more processes I Each sampling step removes 2/3 oftheitemsandadds10moreitems I The nth pass through the data has to process n rows of the table Lecture 2 3/ 16

4 Guess the Next Number ?? ?? ?? ?? ?? Strategy: Look for Patterns I Relate each term to previous terms (arithmetic formula) I Describe in terms of position in sequence I Recognize the set of integers from the examples Lecture 2 4/ 16

5 Patterns Example I Describe the sequence each of the three ways Solution I Relate each term to previous terms Each term is 2 were than the previous first term I I Describe in terms of position in sequence term nth term = 2h I I Recognize the set of integers from the examples Theodd numbers Istarting from D Lecture 2 5/ 16

6 . term 1st More Patterns (Exercise) Problem: For the sequence I Describe it each of the three ways (recursive indexbased recognitionbased) each term is L more than pnev. term is 4 nth term ant 2 even numbers Hati ) starting at 4 I If current term is 898 give next three terms I What is the 1000th term? (Which description is most helpful?) 2002 Lecture 2 6/ 16

7 Sequences and Sequence Notation Recursive Formula Each term is described in relation to previous terms via a recurrence relation Closed Formula Each term is described in terms of its position in the sequence Sequence Notation Sequence name is a lowercase letter (a b...)andasubscript gives position in sequence: a n =nthterminsequencea Example I a = I a 1 = 1 a 2 = 3 a 5 =9 (it s like a function; subscript = ordinal number) I Closed formula: a n =2n 1(forn 1) I Recursive formula: a 1 =1anda n = a n 1 +2(forn 2) Lecture 2 7/ 16

8 Examples For the sequence a n =2 n 1: I Write the first 5 terms: I Value of 10th term: a 10 = I 023 I Formula for (k + 1)st term: I Formula for b i = a 2i 3 Apg : For the sequence a n = a n 1 i i. : In 9= : I a Write the first 5 terms: a 1 da a = It t5 5= I Recursive formula for 80th term: a 80 t = 5 = I = Recursive formula for (k + 1)st term: Anti Ack its Ak +5 " I Recursive formula for a 2j 3 : Gj z Gaj g. t 5 Ajay is Lecture 2 8/ 16

9 . l Discovering Patterns in Sequences Give Recursive and closed formulas: in :* ?? I Papania an=p ?? 8h tninttnn 3. 1 T.it#oWFanaganbnt3=2nt3f " aebntdn.it ?? S O ) " ?? an 2am 9=2 " factorial ?? an I =1 quz.gazez.si same n I recursive inat!g: An " I ' Nan (1) look for di erences and quotients how fast do the numbers grow? (2) compare to simple series with same recurrence 1dL above diet see examples n! Ain Lecture 2 9/ 16 blind "

10 ARockstarSequence:FibonacciNumbers The Sequence The recurrence relation F 1 = F 2 =1andF n = F n 1 + F n 2 for n 3 The closed formula (Binet s formula) F n = 1 p 5 1+ p 5 2! n 1 2 p! n! 5 Applications include (see Fibonacci Quarterly): 1. Fibonacci search Fibonacci heaps 2. Biology and more (leaf/petal patterns tree branching...) Lecture 2 10/ 16

11 D it '... Sh ' Closed Formulas from Recursive Formulas (Can be Hard) Example 1 I a 1 =2anda n =3a n 1 I Compare to simpler sequence b n =3 n I Or write out terms without simplifying a eh Aj 23 93? ". TILNEY bn s anirgbn 35ns. = 2^3 Example 2 I s 1 =1ands n = s n 1 + n It 2+3 Get Si H2 = ' I tht th Sn = I t 2 t. Sme n t F i = t ( n.. t I N month Sn NY Lecture 2 11/ 16 z

12 Recursive Formulas from Closed Formulas (Much Easier) Example 1: Finding a recursive formula I a n =3n +5 I Compute some terms and stare at them (or try algebra) than an _ am e8 Example 2: Verifying a recursive formula I a n =3 n I Show that a n =3a n Amit 4=313 " " 14= ) I 3h 2 = An Lecture 2 12/ 16

13 Exploiting Knowledge About a Sequence Example 1 I a 1 = 11 and a n = a n 1 +5 I a 213 =1071 I What is a 214? An e =2076 Example 2 I a n =2 n 1 I Sum of first 19 terms is I What is the sum of the first 20 terms? [Hint: = ] = It I Lecture 2 13/ 16

14 Sums Notation for sums nx a k = a 1 + a a n = sum of first n terms of sequence a k=1 Extended notation for sums nx a k = a m + a m a n k=m Example: Evaluate the sums " 3X 3X I (2k 1): dg.mn?almei1t3t5 I k 2 : k=1 5=9 k=3 " 4X 3X 4 intention I 3 j 1 : 1+3+9*27481 I k(k + 1) : It son sin j=0 = ht It's 7534 k=1 Lecture 2 14/ 16

15 Aw Sums: More Examples : Notation for sums nx a k = a 1 + a 2 n + + a n = sum of first n terms of sequence a k=1 Examples :. me I Sum of first 10 numbers in sequence a k =1/k with k 1 anew I I I ( 4) + ( 1) Ah t3 Kil Lecture 2 15/ 16

16 Stability of Sequences Example Give the first 4 terms of a n =3a n 1 6with I a 1 = 2: I a 1 = 4: I a 1 = 3: Get =4 6 age I 0 az = 93 AE 3 as = =22 dye 30 04=3 aye a n a n = 0.9 a n a 1 = 1 a 1 = 30 a 1 = n Lecture 2 16/ 16