Fundamentals of Mathematics. Pascal s Triangle An Investigation March 20, 2008 Mario Soster

Transcription

1 Fudmetls of Mthemtics Pscl s Trigle A Ivestigtio Mrch 0, 008 Mrio Soster

2 Historicl Timelie A trigle showig the iomil coefficiets pper i Idi ook i the 0 th cetury I the th cetury Chiese mthemtici Yg Hui presets the rithmetic trigle I the 6 th cetury Itli mthemtici Niccolo Trtgli presets the rithmetic trigle

3 Yg Hui s Trigle

4 Historicl Timelie cot Blsé Pscl 6-66, Frech Mthemtici who pulished his first pper o coics t ge 6, wrote tretise o the rithmeticl trigle which ws med fter him i the 8 th cetury (still kow s Yg Hui s trigle i Chi) Kow s geometric rrgemet tht displys the iomil coefficiets i trigle

5 Pscl s Trigle Wht is the ptter? Wht is the ext row goig to e? We re tkig the sum of the two umers directly ove it.

6 How does this relte to comitios? Usig your clcultor fid the vlue of: Wht ptter do we otice? It follow s Pscl s Trigle

7 So, Pscl s Trigle is: r = 0 = r = = 0 r = = 0 r = = 0

8 Pscl s Idetity/Rule The sum of the previous two terms i the row ove will give us the term elow. r r r

9 Exmple : ) How do you simplify ito sigle expressio? ) How do you write s expded expressio?

10 ) Use Pscl s Idetity: r r r =, d r =

11 ) Use Pscl s Idetity: r r r + =, d r + =, so = d r = Or, wht is? If you sid 9 try i your clcultor: 9 They re the sme thig! Therefore C(,r) is equivlet to C(,-r)

12 Exmple : A former mth studet likes to ply checkers lot. How my wys c the piece show move dow to the ottom?

13 Use Pscl s Trigle: 9 6

14 Exmple : How my differet pths c e followed to spell the word Fudmetls? F U U N N N D D D D A A A A A M M M M M M E E E E E E E N N N N N N T T T T T A A A A L L L S S

15 Use Pscl s Trigle: Therefore there re (6 + 6) = 9 totl wys. Usig comitios, sice there re rows d the fil vlue is i cetrl positio the there C(,6) = 9 totl wys.

16 Exmple : The GO Tri Sttio is locks south d locks est of studet s house. How my differet wys c the studet get to the Go Tri Sttio? The studet c oly go south or est.

17 Drw mp: Studet s House Go Tri Sttio Therefore there re differet wys of goig from the studet s house to the GO Tri sttio. Note: Usig comitios: 0 C((# of rows + # of colums), (# of rows)) C(7,) =

18 Try This: Expd (+) 6

19 Biomil Theorem The coefficiets of this expsio results i Pscl s Trigle The coefficiets of the form re clled iomil coefficiets r r r ) ( r

20 Exmple : Expd (+ )

21 Use the Biomil Theorem: ( ) 0 Wht ptters do we otice? 6 Sum of the expoets i ech sectio will lwys equl the degree of the origil iomil The r vlue i the comitio is the sme s the expoet for the term.

22 Exmple : Expd (x )

23 Use the Biomil Theorem: ) ( ) )( ( ) ( ) ( ) ( ) ( ) ( 0 ) ( x x x x x () ) )( ()( )() (6)( ) )( ()(8 6 x x x x 8 6 x x x x ) ( ) )( ( ) )( ( 6 ) )( ()( x x x x

24 Exmple 6: Express the followig i the form (x+y) 0

25 Check to see if the expressio is iomil expsio: 0 The sum of the expoets for ech term is costt The expoet of the first vrile is decresig s the expoet of the secod vrile is icresig = So the simplified expressio is: ( + )

26 Geerl Term of Biomil Expsio The geerl term i the expsio of (+) is: tr r where r =0,,, r r

27 Exmple 7: Wht is the th term of the iomil expsio of (+)?

28 Apply the geerl term formul! = (+) r = th term wted (r +) = tr t t r r 8 9 r

29 Other Ptters or uses: Fiocci Numers (foud usig the shdow digols ) Figurte Numers Mersee Numer Lucs Numers Ctl Numers Beroulli Numers Trigulr Numers Tetrhedrl Numers Pettope Numers

30 Sources: Grde Dt Mgemet Textooks (check out this wesite, select Eglish) or use