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Vklad novho u iva je spojen s opakovnm zkladnch pojmo z oblasti vlastnost posloupnost. }ci aplikuj zskan poznatky na aritmetick posloupnosti.LqPPh      /8  ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>>  (     6d   `}   XKlepnutm lze upravit styl pYedlohy nadpiso.- -  0i   `   Klepnutm lze upravit styly pYedlohy textu. Druh roveH TYet roveH tvrt roveH Pt roveH,    `  0s  ^ `   >*  0z  ^    @*  0  ^ `   @*  \A޽h ?Balic papr"` 3380___PPT10.-`c/ Vchoz nvrh  (     <Ȃ6 w  < Opakovn:   <L "`,$ 0 >Posloupnost se nazv aritmetick, prv kdy~ existuje takov reln slo d, ~e pro ka~d pYirozen slo n je an+1 = an + d, kde d je DIFERENCE posloupnosti.K   ,p'  <^6 "` ,$ 0 >Posloupnost se nazv rostouc, prv kdy~ pro vaechna pYirozen sla r, s plat: Je-li r < s, pak ar < as.Jme    << "` ,$ 0 @Posloupnost se nazv klesajc, prv kdy~ pro vaechna pYirozen sla r, s plat: Je-li r < s, pak ar > as.Jnf  H  0޽h ? 33___PPT10c.-0wg+/D' = @B DJ' = @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<* 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<* 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<* ++0+   ++0+ ? ++0+ ? +` $R(  $ $ < w ,$ 0 < Opakovn: R $ < "`,$ 0 Posloupnost se nazv shora omezen, prv kdy~ existuje takov reln slo h, ~e pro ka~d pYirozen slo n je an d" h . p|M "r\ $ < "`_ ,$ 0 Posloupnost se nazv zdola omezen, prv kdy~ existuje takov reln slo d, ~e pro ka~d pYirozen slo n je an e" d .|{M " rx $ <_"` U,$ 0 Posloupnost se nazv omezen, prv kdy~ je shora omezen a zdola omezen.LLH $ 0޽h ? 33F>___PPT10.-0wg+D' = @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<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$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<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*$++0+$  ++0+$  ++0+$  ++0+$  + (B(  ( ( <J< w ,$ 0 8loha: ( << ,$ 0 Na rtnte grafy aritmetickch posloupnost, kter jsou dny prvnm lenem a diferenc.XX^ ( 6 ( S A ??"?Y8 $D 0 ( 6A ?}78 $ 0H ( 0޽h ? 33___PPT10.-E8[+0|D' = @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<*(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?\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+k% X3 ,Z(  , , <`;< "`h dU dan posloupnosti ur ete rost, klesn, omezen.33x , << "`H4,$D 0 Posloupnost je rostouc. Zdovodnn: Pro vaechna pYirozen n je an+1 = an + 1, tj. an+1 > an. a<     PB  , <6 "`,$D 0 Posloupnost je zdola omezen. Zdovodnn: Pro vaechna pYirozen sla n je an e" -2. Posloupnost nen shora omezen. Zdovodnn: Ke ka~dmu relnmu slu h najdeme v~dy pYirozen n takov, ~e an > h . Zvr: Posloupnost nen omezen.H N"""" "*)",Nv)H , 0޽h ? 336+.+___PPT10+.-G+?u(~D*' = @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<*,D4' =%(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' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,&a%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,&aD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,&aDn' =%(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' =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' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,,W%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,,WD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,,WD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,Ww%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,WwD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,WwD4' =%(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<*,xD' =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<*,+  D<0(  0^ 0 6 0 C A ?"?8 $D 0 0 6A ?  8 $ 0H 0 0޽h ? 33 { ___PPT10[ .-i+cD ' = @B D ' = @BA?%,( < +O%,( < +Dn' =%(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?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*0+0+(+00+Ip \3 4^(  4 4 <- "`h dU dan posloupnosti ur ete rost, klesn, omezen.33~ 4 <| "`H4,$D 0 Posloupnost je klesajc. Zdovodnn: Pro vaechna pYirozen n je bn+1 = bn  0,5, tj. bn+1 < bn. d=     PC  4 < 6< "`">,$D 0 Posloupnost je shora omezen. Zdovodnn: Pro vaechna pYirozen sla n je bn d" 3. Posloupnost nen zdola omezen. Zdovodnn: Ke ka~dmu relnmu slu d najdeme v~dy pYirozen n takov, ~e bn < d . Zvr: Posloupnost nen omezen.H M"""" "*)",Nu)H 4 0޽h ? 336+.+___PPT10+.-G+zQD*' = @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<*4D4' =%(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<*4'D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4'D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4'd%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4'dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4'dDn' =%(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' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4,D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4,V%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4,VD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4,V      !"#$%&'()*+,-./0143a56789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`cbdefghijklmnopqrstuvwxyz{|}~D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4Vv%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4VvD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4VvD4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4w%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4wD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4wD' =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<*4+ (   8(  8^ 8 6 8 C A  ?"?wt8  $D 0 8 < "` E,$ 0 .Ji~ z grafu plyne, ~e uveden posloupnost nen ani rostouc ani klesajc, ale je omezen. Jedn se o konstantn posloupnost.  8 6A  ?zd  8  $ 0H 8 0޽h ? 33>6___PPT10.-+*D' = @B D=' = @BA?%,( < +O%,( < +Dn' =%(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<* 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<*8+h+0+8m  ++(+ 80+p 0 @<(  < < < w  > Zvr:K < <B "`q%,$ 0 SAritmetick posloupnost s diferenc d je rostouc pro d > 0 a klesajc pro d < 0..T$/[ < << "`k $ 0LD___PPT9&  YPro aritmetickou posloupnost s diferenc d plat: Je-li d > 0, pak je zdola omezen, ale nen shora omezen. Je-li d < 0, pak je shora omezen, ale nen zdola omezen. Je-li d = 0, pak je shora i zdola omezen tj. omezen.F3) 8XH < 0޽h ? 33  ___PPT10 .-+(D ' = @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<*<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<*<+p+0+<m  ++0+<m  ++ ` @&(  @ @ < "`wz ,$ 0 BCvi en: D @ <ĩ< "` ,$ 0 dU dan posloupnosti ur ete rost, klesn, omezen.33^ @ 6 @ C A  ?"?3 Z8  $@ 0 @ < "` %,$ 0 @Xeaen:^  @ 6  @ C A  ?"?; 8 $D 0  @ < "`s G+,$D 0 Posloupnost je aritmetick s diferenc d > 0 . Zvr: Posloupnost je rostouc a zdola omezen0`0(H @ 0޽h ? 33##___PPT10".-sN+zW2D"' = @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<*@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<*@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' =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<* @/D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* @07%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* @07D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* @07D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* @8`%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* @8`D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* @8`++0+@p  ++0+@p  ++0+@m  ++ d\ H(  H H < "`wz ,$ 0 BCvi en: D H <x_"` ,$ 0 dU dan posloupnosti ur ete rost, klesn, omezen.33^ H 6 H <_"` %,$ 0 @Xeaen:^ H 6  H <8_"`s G+,$D 0 Posloupnost je aritmetick s diferenc d < 0 . 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