Discrete Mathematics I Tutorial 12

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Marianna Fox
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1 Discrete Mthemtics I Tutoril Refer to Chpter 4., 4., 4.4. For ech of these sequeces fid recurrece reltio stisfied by this sequece. (The swers re ot uique becuse there re ifiitely my differet recurrece reltios stisfied by y sequece.) ) = 3 b) = c) = d) = + e) = + ( -) f) =! ), 3 b) = -, =, c) d), e) f), ( ),. Show tht the sequece {} is solutio of the recurrece reltio = -l if ) = - +. b) = 5(-l) - +. c) = 3(-) d) = ) if the ( ) 3, ( ) 4 so 9 3 ( 4) 9 b) if 5( ) the 5( ) 3, 5( ) 4 9 5( ) 3 (5( ) 4) 9 5( ) c) if 3( ) the 3( ) 3, 3( ) 4 9 3( ) 3 (3( ) 4) 9 3( ) 7 3, (7 4) 9 7 d) if 7 the Discrete Mthemtics I Tutoril

2 3. ) Fid recurrece reltio for the blce B() owed t the ed of moths o lo t rte of r if pymet P is mde o the lo ech moth. [Hit: Epress B() i terms of B( - ) d ote tht the mothly iterest rte is r /.] b) Determie wht the mothly pymet P should be so tht the lo is pid off fter T moths. r ) B( ) B( ) ( ) P r b) let m the B( ) B( ) m p ( B( ) m p) m p Let = T, If B( T ) m B( ) mp p m ( mb( ) p) mp p... m B() m p... p m m B() p m T T m The m B() p m r T r T ( ) B() m B()( m) So p T m r T ( ) where B() is the mout of origil lo Discrete Mthemtics I Tutoril

3 4. ) Fid recurrece reltio for the umber of bit strigs of legth tht coti pir of cosecutive s. b) Wht re the iitil coditios? c) How my bit strigs of legth seve coti two cosecutive s? ) let deote the umber of bit strigs of legth tht coti pir of cosecutive s. The umber of bit strigs of legth tht coti pir of cosecutive s c be couted s the umber of such strigs of tht strt with plus the umber of such strig s tht hs pir of cosecutive s, the umber is. The umber tht strt with zero c be further broe dow to the umber tht strt plus the umber tht strt, the umber strt with is. Sice every strig tht strt with cotis pir of cosecutive zeros, this gives such strigs, tht is to sy b), c) ( ) ( ) ) Fid recurrece reltio for the umber of wys to lyout wlwy with slte tiles if the tiles re red, gree, or gry, so tht o two red tiles re djcet d tiles of the sme color re cosidered idistiguishble. b) Wht re the iitil coditios for the recurrece reltio i prt ()? c) How my wys re there to lyout pth of seve tiles s described i prt ()? ) b) 3, 8 c) ( ) ( ) ( ) ( ) Discrete Mthemtics I Tutoril 3

4 6. Wht is the geerl form of the solutios of lier homogeeous recurrece reltio if its chrcteristic equtio hs roots,,,, -, -, -, 3, 3, -4? 3 ( ) ( )( ) ( )3 ( 4) Wht is the geerl form of the prticulr solutio gurteed to eist of the lier ohomogeeous recurrece reltio = F() if ) F() = ( - )( -)? b) F() = 4? c) F() =? r r r r r3 r4 8 6,, ) b) c) F ( ) ( )( ) the for s r ( p p p )( ) ( ) ( p) 3 F s s 4 ( ), ( p p p p p ) ( p) F( ), s, s r, s r, p p ( p) Discrete Mthemtics I Tutoril 4

5 8. Solve these recurrece reltios together with the iitil coditios give. ) = - for, = 3 b) = -l for, = c) = 5-l - 6- for, =, l = d) = for, = 6, l = 8 e) + = for, =, l = 8 f) = for 3, = 5, l = -9, d = 5 g) = for, = d l = 4 h) = 4- + ( + 3)4 for, = 8 ) r so, C, C 3 As result, 3 b) r so C C, C As result, c) 5 6 r r r r 5 6,, 3 So C C 3 C C 3 C C C C 3 C 3C C 3, C As result, 3 3 d) 4 4 r r r 4 4, So, ( C d) ( ) 6 d d d c d c ( ) 8, 4, As result, ( d) Discrete Mthemtics I Tutoril 5

6 e) r r r r 4 5,, 5 C C ( 5) C C ( 5) C C C C ( 5) C 5C 8 C 3, C 3 ( 5) f) r r r r 3 3, b c d ( ) ( ) b c d d ( ) ( ) 5 b c b c ( ) ( 5) 9, 4 b c b c b c ( ) (4 5) 5, 4,, 3 As result, ( ) ( 3 5) g) 4 3 r 4r 3 r = or r = 3 ( h) c 3 c F( ) ( 3) p( ) ( b c) ( b c) ( ) ( 4) ( 4) - - 4( )( b( ) c) 3 b b 4 4b 8b c 3 5 b c -4 4 c 4b( ) c 4b 4b 8b c 3 (4b ) ( 8b c 3) 5 4* c c * ( )( b( ) c) 4c( ) 3b( ) 8b 4b 4c 4c 3b 3c( ) 3 3 b b 3c 6c 3 Discrete Mthemtics I Tutoril 6

7 4 c c 5 8 c 3* c 4 39 c c * * 3 h) r 4 4c ( ) h F( ) ( 3) 4 ( b c) 4 ( p) ( b c)4 4( )( b( ) c)4 ( 3) 4 b c b b c b c 3 b b c 3 ( b) ( b c 3) b b c 3 b / c 7 / ( p) 7 ( ) 4 ( p) 8 4c ( h) c 7 ( )4 8 7 ( )4 4c Discrete Mthemtics I Tutoril 7

8 9. Use geertig fuctios to determie the umber of differet wys ideticl blloos c be give to four childre if ech receives t lest two blloos. Becuse we hve ideticl blloos to give four childre d ech oe receives t lest two blloos, so the miml blloos tht oe child could hve is 4. So we oly eed to clculte the coefficiet of i the epsio of: ( ), the coefficiet of i this product is, so there re differet wys.. Use geertig fuctios to fid the umber of wys to choose doze bgels from three vrieties egg, slty d pli if t lest two bgels of ech id but o more th three slty bgels re chose. Becuse we hve bgels to choose, ech id of bgels hve t lest two, so the miml bgels we c te for oe id is 8. So we oly eed to clculte the coefficiet of i the epsio of: ( ) ( ), the coefficiet of i this product is 3, so there re 3 differet wys.. Wht is the geertig fuctio for the sequece {c}, where c is the umber of wys to me chge for dollrs usig $ bills, $ bills, $5 bills, d $ bills? 4 5 f ( ) (...)(...)(...)(...) 5 ( )( )( )( ). Wht is the geertig fuctio for {}, where is the umber of solutios of = whe,, d 3 re itegers with l, 3, d 3 5? the coefficiet of i the epsio f ( ) (...)( )( ) (...)( ) ( ) 4 3 Discrete Mthemtics I Tutoril 8

9 3. Use geertig fuctios to solve the followig recurrece reltio: ) = with iitil coditios = 6 d = 3. b) = -l with iitil coditios = 4 d =. +++ c) = with iitil coditios = d = 5. d) = with iitil coditios =, = 6. () 5 6 Let G( ) G( ) 5 G( ) 6 G( ) (3 5 6) G( ) ( ) 6 5 ( )( 3 ) ( )( 3 ) ( 3 ) ( 3 ) ( 83 ) 83 Discrete Mthemtics I Tutoril 9

10 (b) Let G( ) G( ) ( ) ( ) G G ( ) G( ) 3( ) G( ) ( ( ) ) ( ) ( )( ) ( )( ) ( )( ) ( ) 8 38 ( ) C(, ) ( ) [ ( ) ] Discrete Mthemtics I Tutoril

11 (c) 4 4 let G( ) G( ) ( 4 4 ) ( (4 4 ) ( ( ) ) 4 ( ) G G ( ) 4 ( G( ) ) 4 G( ) 3 ( ) ( ) G ) 4 3 ( ) ( ) ( ) 4 ( ) G( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 ( ) 5 ( ) 3 [ 3 5( ) 3 ( ) 5( ) ( )] ( ) (5 5 )( ) ( ) Discrete Mthemtics I Tutoril

12 (d) let G( ) G( ) ( 3 4 6) ( G( ) ) 3 G( ) ( 3 ) G( ) G( ) ( )( 3 ) ( 4 )( )( 3 ) ( )( )( 3 ) ( ) [ ( ) 4 3 ] ( ) Discrete Mthemtics I Tutoril

PROGRESSIONS AND SERIES

PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.