«A first lesson on Mathematical Induction»

Transcription

1 Bcgou ifotio: «A fist lesso o Mtheticl Iuctio» Mtheticl iuctio is topic i H level Mthetics It is useful i Mtheticl copetitios t ll levels It hs bee coo sight tht stuets c out the poof b theticl iuctio, but e uble to epli wh it is cceptble poof This ticle will be eploig soe pst A level thetics questios to eostte its pplictio Wht is iuctio? Cosie the su of ( ) st te su : st tes su: st tes su : 9 st tes su : Fo this fist sus, we c e guess tht the su hve this ptte: st te su : st tes su: = st tes su : 9 = st tes su : = st tes su : : : The w tht we e this guess fo the obsevtio of few eples, is clle iuctio Bsicll, if the obseve ptte cotiues, the it woul le to the esult of ou guess The ol touble is tht this obseve ptte ees ot cotiue As esult, the guess woul be wog I spite of the fct tht ou guesses fo few eples NOT be coect, this is essetill how theticis iscove the thetics Becuse, ou guess (cojectue) ot be ight, thee is lws ee to pove the guess Pge of 0 A level Mth Note KLAg Jul 0

2 A theticl poof is essetill to show tht the esult is tuth beo oubt Wht is Mtheticl iuctio? A siple cse of this poof begis with: fo the fist cse, sttig cse, it is tuth, suppose it is tuth fo bit th cse, the show tht the ( + )th is lso tuth Sice the fist cse is tuth, th cse tuth les to ( + )th cse is lso tuth, theefoe it is tuth fo ll cses Let s stu this eple cefull, Pove tht =, fo ll positive iteges, Let P() be = Whe =, P(), LHS = RHS = Theefoe, LHS = RHS, P() is tuth Suppose =, P(), is tuth, whee is positive itege = Whe = +, P( + ), ( + ) = + ( + ) = P( + ) is tuth Sice P() is tuth, if P() is tuth, the P( +) is tuth, so P() is tuth fo ll positive iteges Pge of 0 A level Mth Note KLAg Jul 0

3 Wh it wos? Let te close loo t the eple: To pove tht =, fo ll positive iteges, Let P() be = Whe =, P(), LHS = RHS = Suppose =, P(), is tuth, = Requie to pove tht P( + ) is, ( + ) = Theefoe, LHS = RHS, P() is tuth, whee Copig P() P( +), the ol iffeece is the itiol te ( + ) i the seies Whe = +, P( + ), LHS = ( + ) = = + ( + ), eplce the st tes with = RHS P( + ) is tuth Sice P() is tuth, if P() is tuth, the P( +) is tuth, so P() is tuth fo ll positive iteges It wos becuse, P() begis with P() Sice we ow tht P() is tuth, the b the iuctive step, it follows tht P() is lso tuth Agi, P() is tuth, b the iuctive step, P() is tuth s well B the epetitio of pplig the iuctive step (if P() is tuth, the P( +) is tuth), P() is tuth fo ll positive itege Pge of 0 A level Mth Note KLAg Jul 0

4 The listig of P( + ) the pt o equie to pove e useful to show the iffeece betwee P( ) P( + ), wht the e esult shoul be These e tpicll oitte i st witig Let s begis with few woe eples fo the pst A level eitios: Eple : Solutio: Pove b iuctio tht Let P() be [] Whe =, P(), LHS = RHS = 0 Theefoe, LHS = RHS, P() is tuth Suppose =, P(), is tuth,, whee Requie to pove tht P( + ) is, Whe = +, P( + ),, eplce the st tes with P( + ) is tuth Sice P() is tuth, if P() is tuth, the P( +) is tuth, so P() is tuth fo ll positive iteges Reebe, ust use P() i the povig of P( + ) Pge of 0 A level Mth Note KLAg Jul 0

5 Pge of 0 A level Mth Note KLAg Jul 0 Eple : (i) Pove, b iuctio o othewise, tht Hece show tht [] (ii) A sequece of positive iteges u, u, u, is efie b, u = u + = u + fo ; thus the fist few tes of the sequece e,,,, Pove b iuctio tht u = ( ) [] Solutio(i): Let P() be Whe =, P(), LHS = RHS = Theefoe, LHS = RHS, P() is tuth Suppose =, P(), is tuth,, whee Requie to pove tht P( + ) is, o ltetivel, Whe = +, P( + ), P( + ) P() =

6 P( + ) is tuth Sice P() is tuth, if P() is tuth, the P( +) is tuth, so P() is tuth fo ll positive iteges Solutio(ii): Let P() be u = ( ) Whe =, P(), u =, give u = Theefoe, P() is tuth Suppose =, P(), is tuth, u = ( ), whee Requie to pove tht P( + ) is, u + = ( ) Whe = +, P( + ), u + = u + =( ( ) )+ =( ) P( + ) is tuth Sice P() is tuth, if P() is tuth, the P( +) is tuth, so P() is tuth fo ll positive iteges Eple : Pove b theticl iuctio tht, fo ll o-egtive iteges, is ivisible [] Solutio: Pge of 0 A level Mth Note KLAg Jul 0

7 Let P() be be ivisible b Whe = 0, P(0), P(0) is ultiple of Theefoe, P(0) is tuth Suppose =, P(), is tuth,, whee 0, is positive itege Requie to pove tht P( + ) is, p whee p is positive itege Altetivel, P( + ) P() = p = q whee q is positive itege Whe = +, P( + ), P( + ) P() = 8 Sice is ultiple of, theefoe P( + ) is tuth Sice P(0) is tuth, if P() is tuth, the P( +) is tuth, so P() is tuth fo ll o-egtive iteges Eple : It is give tht f (i) Show tht f f 8 (ii) Solutio(i): [] Hece, o othewise, pove b theticl iuctio tht f is ivisible b fo eve positive itege [] Give tht f, the f Pge of 0 A level Mth Note KLAg Jul 0

8 Pge 8 of 0 A level Mth Note KLAg Jul 0 8 f f Solutio(ii): Let P() be f tht is ivisible b Whe =, P(), 8 f P() is ultiple of Theefoe, P() is tuth Suppose =, P(), is tuth, f, whee, is positive itege Requie to pove tht P( + ) is, p f whee p is positive itege Altetivel, P( + ) + P() = q whee q is positive itege Whe = +, P( + ), 8 f f Sice f f f e ultiple of, the f is lso ultiple of Theefoe P( + ) is tuth Sice P() is tuth, if P() is tuth, the P( +) is tuth, so P() is tuth fo ll positive iteges Eple : The sequece,,, is such tht = fo =,,, Pove b iuctio tht > fo ll [] Solutio: Let P() be >

9 Whe =, P(),, theefoe, P() is tuth Suppose =, P(), is tuth,, whee Requie to pove tht P( + ) is, o 0 Whe = +, P( + ), 8 Whe, -½ Theefoe 0, P( + ) is tuth Sice P() is tuth, if P() is tuth, the P( +) is tuth, so P() is tuth fo ll positive iteges Eple : Give tht si, fi, siplifig ou esults s f s possible, show tht si cos [] Use iuctio to estblish epessio fo, whee is positive itege [] Solutio: Pge 9 of 0 A level Mth Note KLAg Jul 0

10 Pge 0 of 0 A level Mth Note KLAg Jul 0 cos si cos si cos si Let P() be cos si whee si Whe =, P(), Fo eivtive, cos si Fo P(), cos si Theefoe, P() is tuth Suppose =, P(), is tuth, cos si, whee Requie to pove tht P( + ) is, cos si Whe = +, P( + ), cos si, eplce the cos si cos si cos cos si cos cos si

11 si cos si cos P( + ) is tuth Sice P() is tuth, if P() is tuth, the P( +) is tuth, so P() is tuth fo ll positive iteges Eple : It is give tht l l b, whee b epe ol o (i) Fi, [] (ii) Use theticl iuctio to estblish foul fo [] Solutio(i): Let l, l Theefoe, l b l Theefoe, l Theefoe, Solutio(ii): l b l b l l b whee Let P() be Whe =, P(), Pge of 0 A level Mth Note KLAg Jul 0

12 Pge of 0 A level Mth Note KLAg Jul 0 Fo st eivtive, Fo P(), Theefoe, P() is tuth Suppose =, P(), is tuth,, whee Requie to pove tht P( + ) is, o Whe = +, P( + ), l l l l b b, eplce l l b l b Theefoe P( + ) is tuth Sice P() is tuth, if P() is tuth, the P( +) is tuth, so P() is tuth fo ll positive iteges Fo both the eple, it is ost ipotce fo us to guess the coect foule This iuctive guess chec is cetl i thetici s wo As show i the bove eple, the su to the st th tes foule e give i ech questio The e questio is tht how c we eive t these foule Fo eple, fi the su of the st th tes this seies :

13 If this ptte cotiues, iuctivel, the su to the st th tes The followig eples will eostte soe of the ws to eive t this su of the st th tes foul Eple 8: Deive the su of this seies ( ) i te of Let the st te, u ; te, u ; te, u ; th te, u ; th te, u The ube sequece,,,,,,, fo Aithetic Pogessio, AP, s thee is coo iffeece,, of betwee two successive tes I geel, AP c be epesse s,,,,, u ; u ; u ; u ;, u Let the su of the seies of the st tes be S S () Rege the tes i S, S () A () (), S S S S I this seies,,, S so,, this is the AP su foul, this esult is the se s the oe i the iuctio Pge of 0 A level Mth Note KLAg Jul 0

14 ( ) The eivtio of this foul is NOT the poof of the foul! You still ee to ppl the theticl iuctio to pove its coectess B the w, AP c be epesse i ecusive w s : u u u, Eple 9: Deive the su of this seies i te of Let the st te, u ; te, u ; te, u 9 ; th te, u ; th te, u The ube sequece,,, 9,,,, fo Geoetic Pogessio, GP, s thee is coo tio,, of betwee two successive tes I geel, GP c be epesse s,,,,, u ; u ; u ; u ;, u Let the su of the seies of the st tes be S S () Multipl i S, S () Subtct () fo (), S S, this is the GP su foul, fo I this seies,,, S So, Pge of 0 A level Mth Note KLAg Jul 0

15 Agi, GP c be epesse i ecusive w s : u u u, We c the ppl theticl iuctio to pove this esult Eple 0: Deive the su of this seies i te of Let the su of the seies of the st tes be S S () Notice tht this seies cotis coefficiets i AP i GP Multipl i S, S () Subtct () fo (), S S S S I geel, whe te, siil w C, of seies whee C is AP is GP c be su i Eple : Fi the su up to tes of this seies!! Let the su of the seies of the st tes be S S!!!!!!!!!!!!!!!! Pge of 0 A level Mth Note KLAg Jul 0

16 : :!!!!!!!! Suig ll the tes, S!!!!!!!!!!!!!!!!!!!!!!!!! S! S!! S!!!!!! Notice tht ech te is split ito two tes The techique is clle etho of iffeece Eple : Fi the su up to tes of this seies Let the su of the seies of the st tes be S S B ptil fctio, A B C A B C Pge of 0 A level Mth Note KLAg Jul 0

17 Pge of 0 A level Mth Note KLAg Jul : Suig ll the tes, S S S S S Ptil fctio is useful w to ecopose the th te Eple : Fi the su up to tes of this seies

18 Let the su of the seies of the st tes be S S Sice S S S, Usig st suig foule is ipott ppoch Eple : Fi the su up to tes of this seies 0 8 Let the su of the seies of the st tes be S S 0 8 C A coo tio, is peset But the othe fcto C will ee futhe ivestigtio C 0 8 C C So, we hve Pge 8 of 0 A level Mth Note KLAg Jul 0

19 Pge 9 of 0 A level Mth Note KLAg Jul 0 C 0 C 0 C 0 0 C 8 0 C 0 C 0 C C ltetivel, it is possible to guess out this s well 8 0 S S, su of GP seies S 8 0 I coclusio: The st pt of the ticle eplis wh how of the Piciple of Mtheticl Iuctio The few siple eples e seve to illustte its pplictio The pt of the ticle is

20 to eostte soe of the ethos i obtiig the su of th te foul Of couse, stuet shoul le to guess chec the th te, which ot be etiel es Pge 0 of 0 A level Mth Note KLAg Jul 0