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( <pH},$ 0 vDU~it ve stYedoakolsk matematice:## ( <HCT,$D 0  Sou et prvnch n leno ur it posloupnosti Z rekurentnho vyjdYen dokzat sprvnost vzorce pro n  t len Dokaz dlitelnosti danho vrazu pYirozenm slem nebo vrazem Dokazy slovnch loh .-jaH ( 0޽h ? 33___PPT10.#U+ڱCD' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(,Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(,r%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(,rD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(,rDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(r%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(rD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(rDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(+p+0+(H ++0+(H +?  ,(  , , <+H' fDokaz matematickou indukc je veden ve dvou krocch44 , <;H/' |$D 0H@___PPT9" Dok~eme, ~e V(n) plat pro n = 1 (nebo nejmena mo~n pYpustn n) 2. Vyslovme pYedpoklad, ~e tvrzen plat pro libovoln pYirozen slo k. Na jeho zklad dok~eme: Jestli~e plat V(k), pak plat V(k+1)..""8X , <d7H) /I ,$ 0 > Matematicky: ` , c $A ??, d  ^  , 6++ , S A ??"? ] 8 $D 0H , 0޽h ? 33___PPT10.#{+Da' = @B D' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,"%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,"D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,"Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,"E%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,"ED' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,"EDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,F%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,FD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,FD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*,Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*,+8+0+,H +K0   0e (  0 0 <HUl u :loha 1: R 0 <dH"`e k,$ 0 fDoka~te: Pro vaechna pYirozen sla n plat vztah:"4 +^ 0 6 0 S A ??#" `?'8 $D 0 0 <dH"`,$ 0 41.^  0 6  0 S A ??"?S  8 $D 0^ 0 6 0 S A  ??"?m}8 $D 02 0 0IH"`5 ,$ 0 ^2. PYedpoklad: Pro libovoln k pYirozen plat:00 0 <$H"`$ D ,$ 0 = Dokazujeme: ^ 0 6 0 S A ??"? w8 $D  0H 0 0޽h ? 33&&___PPT10%.#lr+}D$' = @B D$' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* 0D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*0Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*0++0+0H ++0+0H ++0+0H ++0+0H +8  4>(  4^ 4 6 4 S A  ??#" `?8 $D 0^ 4 6 4 S A  ??#" `?Tw8 $D 0^  4 6 4 S A   ??#" `?T+ 8  $D 0  4 <H6 V ,$ 0 > Zvr:  4 <H ,$ 0 b0Tvrzen plat pro nejmena mo~n n. Plat-li pro libovoln pYirozen slo k, plat i pro nsledujc. Tedy tvrzen plat pro vaechna pYirozen sla n.H 4 0޽h ? 332*___PPT10 .#5 T+WIDn' = @B D)' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* 4D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* 4D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* 4D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* 4+p+0+ 4H ++0+ 4H +BA %  8 (  8 8 <HHU u FCvi en 1: R 8 <H"`e k,$ 0 fDoka~te: Pro vaechna pYirozen sla n plat vztah:"4 + 8 S A   ??#" `?z' 8  $D 0 8 <(H"`2,$ 0 41.^  8 6^  8 6++  8 S A ??#" `? 98 $D 02 8 0H"`"B,$ 0 ^2. PYedpoklad: Pro libovoln k pYirozen plat:00^ 8 6(( 8 S A  ??#" `?`' 8  $D 0 8 <LH"` ,$ 0 = Dokazujeme: ^ 8 6(( 8 S A ??#" `? ' 8 $D  0^ 8 6(( 8 S A ??#" `?F T 8 $D  0^ 8 6++ 8 S A ??#" `? 8 $D  0 8 <H {,$  0 v6Zvr: Viz pYedchoz loha$H 8 0޽h ? 3333___PPT103.#?2-+gPDY2' = @B D2' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* 8%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* 8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* 8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*8D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*8++0+8H ++0+8H ++0+8o ++0+8o ++0+8o +K;   @<y (  < < <R|Ul u :loha 2: F < <Z|"`e k,$ 0 ZDoka~te: Pro vaechna pYirozen sla n plat:". %^ < 6 < S A ??#" `?<8 $D 0 < <]|"`,$ 0 41.^  < 6  < S A ??#" `?7 (8 $D 02  < 0 c|"`,$ 0 ^2. PYedpoklad: Pro libovoln k pYirozen plat:00  < S A ??"?8 $D 0 < <Lg|"` ,$ 0 = Dokazujeme: ^ < 6^ < 6S S  < S A ??#" `? B 8 $D  0^ < 6q < S A ??#" `?F ' 8 $D  0^ < 6% %  < S A ??#" `?f ' 8 $D  0H < 0޽h ? 33 //___PPT10..#C]\+1 ZD-' = @B D-' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* <%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* <D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* <D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* <%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* <D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* <Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* <%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* <D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* <D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*<++0+<| ++0+<| ++0+ <| ++0+<| +Q `@k(  @ @ <do"`W,$ 0 7Vraz^ @ 6(( @ S A ??#" `?w*8 $D 0. @ < o"`~/,$ 0 NJe dliteln slem 6 podle pYedpokladu(( @ < o"`W,$ 0 7Vraz^  @ 6((  @ S A ??#" `?zF 8 $D 0  @ 0`o"`m],$ 0 \* je dliteln tYemi a  @ <\o"`2BR,$ 0 7vraz^ @ 6((  @ S A  ??#" `?  p8  $D  0D @ 0Xo"`i ,$  0 p je sud pro vaechna pYirozen sla k (je dliteln 2)99 @ <#o"`A `a ,$  0 PTzn., ~e vraz  @ S A !??#" `? 9 s 8 !$D  0 @ <(o ~ 1( ,$  0 ^,je dliteln slem 6. @ <-o M  ,$ 0 > Zvr:0 @ </o"` v,$ 0 PCel uveden vraz je dliteln slem 6))H @ 0޽h ? 33BxB___PPT10XB.!#s+}hD?' = @B D?' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* @D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* @D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* @D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* @D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* @D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* @Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* @%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* @D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* @D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*@D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*@D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*@D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*@+0+0+@o ++0+@o ++0+@o ++0+ @o ++0+ @o ++0+@o ++0+@o ++0+@o ++0+@o ++0+@o + Zvr: D < ro ' *$ D <vo"` <,$  0 vDZcela analogick k pYedchoz loze##H D 0޽h ? 3333___PPT103.##`n+8DY2' = @B D2' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* D%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* DD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*DD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D++0+Do ++0+ Do ++0+Do ++0+Do ++0+Do +1D H (  H H <oUl u :loha 3:  H <oZ z,$ 0 p>Posloupnost je dna rekurentn: ^ H 6 H S A *??"?P 8 *$D 0> H <ęo ?,$ 0 JDoka~te jej vzorec pro n  t len: &&^  H 6  H S A# +??"? 8 +$D 0^  H 6  H S A$ ,??"?]qQ8 ,$D 0,  H 0o"`}^,$ 0 XPYedpoklad: Pro libovoln k pYirozen plat:--^ H 6 H S A& /??"?cB 8 /$D 0 H <(o"`Cc ,$  0 = Dokazujeme: ^ H 6 H S A% 0??"?g 8 0$D  0^ H 6 H S A' 1??"?  8 1$D  0 H <$o z ,$  0 > Zvr:@ H <ܭo"`,$  0 `Dan vzorec plat pro vaechna pYirozen sla n.11H H 0޽h ? 3355___PPT105.&#Sr+TD4' = @B D3' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*HDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*HDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*HD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* HDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* HDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* HD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*HDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*HD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*HDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*HDN' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*HD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*H+P+0+Ho ++0+Ho ++0+ Ho ++0+Ho ++0+Ho ++0+Ho +I )!L(  L L <oU u FCvi en 3:  L <oZ z,$ 0 p>Posloupnost je dna rekurentn: ^ L 6(( L S A( W??"?, 8 W$D 0> L <Po ?,$ 0 JDoka~te jej vzorec pro n  t len: &&^  L 6%%  L S A) X??"?3 g^8 X$D 0^  L 6((  L S A* Y??"?qw8 Y$D 0,  L 0o"`}c,$ 0 XPYedpoklad: Pro libovoln k pYirozen plat:--^ L 6%% L S A+ Z??"?"8 Z$D 0 L <Xo"`5 ,$  0 = Dokazujeme: ^ L 6 L S A. ??"?qz 8 $D  0^ L 6%% L S A- \??"? 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T # 6"`-! 9I  T # <"`ma :2   T # B"`-u I  :II  T # |G"`=! 93  T # L"`   ;III  T # P"`M :1   T # xU"` :IV  T <V"`~ ,$ 0 CSituace pro k = 3x T <l\   z T <P` ,$D 0 h Rovina je k pYmkami rozdlena na pk st Veme (k+1)  n pYmku Tato pYmka prochz (k+1) stmi roviny Ka~dou z nich rozdl na dv sti Vznikne tedy dalach (k+1) st4-$ #H T 0޽h ? 3311___PPT101.6#+RKlD0' = @B D0' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* T%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* TDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*T,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*T,Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T,E%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*T,ED' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*T,EDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*TEo%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TEoD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*TEoDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*To%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*ToD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*ToDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*T++0+T ++0+T ++0+T +!  X(  X X S A3 c??"?]8 c$D 0 X <"` ,$ 0 :Je tedy: ^ X 6(( X S A4 d??"?<T8 d$D 0^  X 6 X S A5 e??"?2 8 e$D 0  X < z 8 ,$ 0 > Zvr:  X <f  ,$ 0 LDan tvrzen bylo dokznoH X 0޽h ? 33___PPT10e.8#@sg+B#D' = @B DL' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*XD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*XDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*XDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*XDN' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* XD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* XD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* XD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* X++0+X ++0+ X ++0+ X +S jb `(  ` ` <=p  TPOU}IT ZDROJE: ` <xNH@8___PPT9 doc. RNDr. ODVRKO, O., DrSc.: Matematika pro gymnzia  posloupnosti a Yady, 1. vydn Praha Prometheus, spol. s r. o. 1995. 126 s. ISBN 80-85849-91-7 PETKOV, J. , MATEMATIKA  pYprava k maturit a pYijmacm zkouakm na vysok akoly, 1. vydn Praha Prometheus, spol. s r. o. 1998, 303 s. 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