\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\froman\fprq2 Times New Roman;}{\f4\fswiss\fprq2 Arial;}{\f5\fmodern\fprq1 Courier New;}{\f6\fswiss\fprq2 Helvetica;}} {\colortbl\red0\green0\blue0;\red0\green0\blue255;\red255\green255\blue255;\red128\green0\blue0;\red255\green0\blue0;\red0\green128\blue0;} \deflang1031\pard\ri4\plain\f5\fs20\cf0\b ________________________________________________________________________________ \par \par Inhalt....: Das Prinzip der vollst\'e4ndigen Induktion \par Kategorie.: Unterrichtsmaterial \par Mathematik: Analysis \par MuPAD.....: 3.0.0 \par Datum.....: 2003-03-04 \par Autoren...: Kai Gehrs \par Funktionen: subs, expand \par ________________________________________________________________________________ \par \plain\f4\fs28\cf0 \par \plain\f4\fs36\cf0\b Das Prinzip der vollst\'e4ndigen Induktion \par \plain\f4\fs28\cf0 \par \plain\f4\fs24\cf5 Dieses Notebook bietet eine kurze, kompakte Einf\'fchrung in das Beweisprinzip der voll- \par st\'e4ndigen Induktion. Am Beispiel des Domino-Day wird das Konzept von Induktionsanfang \par (der erste Stein, der umkippt), Induktionsvoraussetzung (der n-te Stein kippt um) und dem \par Induktionsschlu\'df (wenn der n-te Stein umkippt, so zeige, dass auch der (n+1)-te Stein um- \par kippen wird) erkl\'e4rt. Anschlie\'dfend werden zwei formale Induktionsbeweise mit MuPAD \par diskutiert. Dabei sind - neben den Funktionen subs und expand und einigen elementaren \par Kenntnissen zur Bedienung von MuPAD - keinerlei weitere Voraussetzungen n\'f6tig. \par \plain\f4\fs24\cf0 \par \par \plain\f4\fs24\cf3\b Motivation und Konzept des Arbeitsblatts \par \plain\f4\fs24\cf3 \par Bei der Einf\'fchrung der Integralrechnung \'fcber Rechtecksummen tauchen in der Regel \par unweigerlich spezielle Summenformeln wie z.B. \par \par \pard\li500\ri4\plain\f4\fs24\cf3 {\pict\wmetafile8\picw12174\pich1480\picscalex99\picscaley99\picwgoal6966\pichgoal843 010009000003E70800000B001C0000000000050000000B0200000000050000000C02C8058E2F03 0000001E00050000000C02D0050130050000000B0200000000030000001E00050000000C02D905 0B30050000000B0200000000050000000B0200000000030000001E00050000000C02E2057F3005 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 0C02EB058930050000000B0200000000050000000B0200000000050000000B0200000000050000 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FF04000000F001000004000000F001010004000000F001020004000000F001030004000000F001 040004000000F001050004000000F001060004000000F001070004000000F00108000400000027 01FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF04000000 2701FFFF040000002701FFFF040000002701FFFF040000002701FFFF030000000000 }\plain\f4\fs24\cf3 \par \par \pard\ri4\plain\f4\fs24\cf3 auf: Oft wird etwa die Normalparabel \plain\f3\fs24\cf3\i f \plain\f3\fs24\cf3 (\plain\f3\fs24\cf3\i x\plain\f3\fs24\cf3 )\plain\f3\fs24\cf3\i = x\plain\f3\fs24\cf3 \'b2\plain\f4\fs24\cf3 zur Einf\'fchrung der Riemann-Integration \par im Unterricht benutzt. Man w\'e4hlt sich z.B. das Intervall von \plain\f3\fs24\cf3 0 \plain\f4\fs24\cf3 bis\plain\f3\fs24\cf3 1 \plain\f4\fs24\cf3 und beginnt es suk- \par zessive in mehr gleich lange Teilintervalle zu unterteilen. Beim \'dcbergang zu einem \par allgemeinen Summationsindex steht man im Fall der Normalparabel vor dem Pro- \par blem, die Summe der Quadratzahlen von \plain\f3\fs24\cf3 1\plain\f4\fs24\cf3 bis \plain\f3\fs24\cf3\i n\plain\f4\fs24\cf3 berechnen zu m\'fcssen und an- \par schlie\'dfend den Grenz\'fcbergang f\'fcr \plain\f3\fs24\cf3\i n\plain\f4\fs24\cf3 gegen Unendlich vollziehen zu k\'f6nnen. In der \par Regel bleibt im Unterricht nicht die Zeit, diese Formeln herzuleiten, geschweige \par denn sie mittels vollst\'e4ndiger Induktion zu beweisen. \par \par Dieses Arbeitsblatt soll Abhilfe schaffen. Die Sch\'fcler/innen k\'f6nnen sich selbst- \par st\'e4ndig mit den unten aufgef\'fchrten Inhalten besch\'e4ftigen. Das Prinzip der voll- \par st\'e4ndigen Induktion kann auf diese Weise im Rahmen einer Stunde bzw. einer \par Doppelstunde (je nach Belieben) im Unterricht trotz Zeitmangels vorgestellt wer- \par den. \par \par \plain\f4\fs24\cf3\b Unterrichtliche Voraussetzungen f\'fcr diese Einheit: \par \plain\f4\fs24\cf3 \par \pard\li500\ri4\plain\f4\fs24\cf3 - die Sch\'fcler/innen haben den elementaren Umgang mit MuPAD \par bereits erlernt \par \par - die Berechnung von Summen bis zu einem allgemeinen Index \par wurde zuvor im Unterricht (z.B. im Kontext der Integration) motiviert \par \par - je nach Geschmack kann zuvor eine kurze Erl\'e4uterung des gedank- \par lichen Prinzips der vollst\'e4ndigen Induktion erfolgt sein \par \pard\ri4\plain\f4\fs24\cf3 \par Letzteres sollte nicht zwingend n\'f6tig sein: Das Prinzip der vollst\'e4ndigen Induktion \par wird im folgenden anschaulich am Beispiel des "Domino-Days" verdeutlicht. \par \plain\f4\fs28\cf3 \par \plain\f4\fs28\cf0 \par \plain\f4\fs28\cf0\b Motivation: Das gedankliche Prinzip der vollst\'e4ndigen Induktion \par \tab \tab am Beispiel des "Domino-Day" \par \plain\f4\fs28\cf0 \par Wir alle kennen den "Domino-Day" aus dem Fernsehen, wo ein Team von \par begeisterten Domino-Fans ganze Kunstwerke aus Domino-Steinen errichtet. \par Die Steine werden dabei so geschickt angeordnet, dass an einer ausge- \par zeichneten Stelle zu Beginn nur ein einzelner Stein umgekippt werden muss. \par Dieser l\'f6st dann eine Kettenreaktion aus: Der erste Stein kippt gegen seinen \par Nachbarn, der Nachbar kippt wiederum gegen seinen Nachbarn, der wiederum \par gegen den n\'e4chsten Nachbarn usw. Auf diese Weise fallen alle Steine nach- \par einander um. \par \par Stellen wir uns vor, wir sollten im Voraus begr\'fcnden, warum tats\'e4chlich \plain\f4\fs28\cf0\b alle\plain\f4\fs28\cf0 \par Domino-Steine nacheinander umfallen. Wie w\'fcrden wir dieses Problem am \par geschicktesten l\'f6sen? \par \par Nun: Eine M\'f6glichkeit sicherlich darin, das gesamte Feld von Domino-Steinen \par genaustens in Augenmerk zu nehmen und zu kontrollieren, ob alle Steine in der \par Tat so angeordnet sind, dass der Vorg\'e4nger immer von seinen Nachfolger um- \par sto\'dfen wird. \par \par Wer allerdings schon einmal den "Domino-Day" im Fernsehen gesehen hat, \par der wei\'df, dass man wahrscheinlich Tage brauchen w\'fcrde, bis man davon \par \'fcberzeugt sein kann, dass in der Tat alle Steine umfallen werden. \par \par Geschickter ist die folgende simple Argumentation: Wir wissen, dass der \par erste Stein per Hand umgesto\'dfen wird. Stellen wir uns die Domino-Steine \par einmal mit den Zahlen \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 , \plain\f3\fs28\cf1 2\plain\f4\fs28\cf0 , \plain\f3\fs28\cf1 3\plain\f4\fs28\cf0 , . . . ,\plain\f4\fs28\cf0\i \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 , \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 \plain\f3\fs28\cf1 +\plain\f4\fs28\cf0 \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 , . . . usw. durchnummeriert vor. \par Jetzt nehmen wir einfach an, der Stein mit der Nummer n w\'fcrde in der Tat \par umfallen. Dann kontrollieren wir, ob der Stein \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 nah genug an seinem Nach- \par folger, d.h. dem Stein mit der Nummer \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 \plain\f3\fs28\cf1 +\plain\f4\fs28\cf0 \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 , steht. Ist dies der Fall, so wird \par der Stein mit der Nummer \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 \plain\f3\fs28\cf1 +\plain\f4\fs28\cf0 \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 umfallen. \par \par Sind wir also in der Lage, die obige Argumentation f\'fcr einen beliebigen \par Stein \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 und seinen Nachfolger mit der Nummer \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 \plain\f3\fs28\cf1 +\plain\f4\fs28\cf0 \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 durchzuf\'fchren, so \par k\'f6nnen wir also unser Gegen\'fcber sicherlich davon \'fcberzeugen, dass alle \par Steine umfallen. \par \par Und was hat das alles mit Mathematik zu tun? \par \par Eine ganze Menge! \par \par \par \plain\f4\fs28\cf0\b Formalisierung des Prinzips der vollst\'e4ndigen Induktion am \par Beispiel der Summe der ersten \plain\f3\fs32\cf1\i n\plain\f3\fs28\cf1\b\i \plain\f4\fs28\cf0\b Quadratzahlen \par \plain\f4\fs28\cf0 \par Was wir soeben informell erarbeitet haben, nennt sich das Prinzip der \par vollst\'e4ndigen Induktion. Dieses Prinzip ist eines der wichtigsten in der \par Mathematik und der Informatik. Wir werden jetzt sehen, wie man es \par nutzen kann, um "Aussagen \'fcber den nat\'fcrlichen Zahlen" zu beweisen. \par \par \plain\f4\fs28\cf0\b Aufgabe: \plain\f4\fs28\cf0 Beweise, dass die Summe der Quadratzahlen von \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 bis zu einer \par beliebigen Zahl \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 gegeben ist durch: \par \par \pard\li2500\ri4\plain\f4\fs28\cf0 {\pict\wmetafile8\picw6129\pich1513\picscalex99\picscaley99\picwgoal3483\pichgoal858 010009000003A90500000B001C0000000000050000000B0200000000050000000C02E905F11703 0000001E00050000000C02EB050118050000000B0200000000030000001E00050000000C02FA05 3118050000000B0200000000050000000B0200000000030000001E00050000000C020C06411805 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 0C021C067118050000000B0200000000050000000B0200000000050000000B0200000000050000 000B0200000000030000001E00050000000C022E068118050000000B0200000000050000000B02 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}\plain\f4\fs28\cf0 \par \pard\ri4\plain\f4\fs28\cf0 \par Dabei bedeutet das Zeichen auf der linken Seite der Gleichung \par nichts anderes als: Bilde die Summe der ersten \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 Quadratzahlen. \par \pard\li3000\ri4\plain\f4\fs28\cf0 \par \pard\ri4\plain\f4\fs28\cf0 Wir k\'f6nnen nat\'fcrlich nicht f\'fcr jede nat\'fcrliche Zahl \plain\f3\fs28\cf1\i n \plain\f4\fs28\cf0 ausprobieren, ob die obige \par Formel korrekt ist. Stattdessen gehen wir einfach so vor, wie wir es oben bei \par dem Beispiel des "Domino-Days" getan haben. \par \par \plain\f4\fs28\cf0\b Induktionsanfang: \par \plain\f4\fs28\cf0 \par Wir m\'fcssen sicherstellen, dass der erste Domino-Stein umgekippt wird. Andern- \par falls haben alle anderen keine Chance auch umzufallen. In unserer Situation ent- \par spricht das "Umkippen des ersten Stein" der Probe, ob die obige Formel f\'fcr den \par "trivialen Fall" \plain\f3\fs28\cf1\i n = \plain\f3\fs28\cf1 1\plain\f3\fs28\cf1\i \plain\f4\fs28\cf0 erf\'fcllt ist. Die Summe der ersten \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 Quadratzahlen schrumpft f\'fcr \par den Fall \plain\f3\fs28\cf1\i n = \plain\f3\fs28\cf1 1 \plain\f4\fs28\cf0 auf \plain\f3\fs28\cf1 1\plain\f3\fs19\cf1\up14 2 \plain\f3\fs28\cf1\i = \plain\f3\fs28\cf1 1 \plain\f4\fs28\cf0 zusammen. Wir m\'fcssen also nur noch \'fcberpr\'fcfen, ob \par auch die rechte Seite der Formel \par \pard\li2500\ri4\plain\f4\fs28\cf0 {\pict\wmetafile8\picw6120\pich1552\picscalex99\picscaley99\picwgoal3483\pichgoal885 010009000003A90500000B001C0000000000050000000B0200000000050000000C021006E81703 0000001E00050000000C021A060118050000000B0200000000030000001E00050000000C022106 2818050000000B0200000000050000000B0200000000030000001E00050000000C022C06411805 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 0C0232066818050000000B0200000000050000000B0200000000050000000B0200000000050000 000B0200000000030000001E00050000000C023E068118050000000B0200000000050000000B02 00000000050000000B0200000000050000000B0200000000050000000B0200000000030000001E 00050000000C024506AA18050000000B0200000000050000000B0200000000050000000B020000 0000050000000B0200000000050000000B0200000000050000000B0200000000030000001E0005 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Dies nennt man den \plain\f4\fs28\cf0\b Induktionsanfang\plain\f4\fs28\cf0 . \par Dazu definieren wir die rechte Seite zuerst f\'fcr allgemeines \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf4 {\pntext\f1\'b7\tab}Quadrate_bis_n:= 1/6 * n * (n + 1) * (2*n + 1) \par \pard\ri4\plain\f4\fs28\cf0 \par und setzen dann f\'fcr n den Wert 1 ein: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf4 {\pntext\f1\'b7\tab}subs(Quadrate_bis_n, n = 1) \par \pard\ri4\plain\f4\fs28\cf0 \par Das ist genau der Wert, den wir haben wollten. Wir haben also gezeigt, dass \par die obigen Formel im Fall \plain\f3\fs28\cf1\i n = \plain\f3\fs28\cf1 1 \plain\f4\fs28\cf0 richtig ist oder mit anderen Worten: Der erste \par Domino-Stein ist umgefallen. \par \par \plain\f4\fs28\cf0\b Induktionsvoraussetzung und Induktionsschluss: \par \plain\f4\fs28\cf0 \par Jetzt gehen wir davon aus, dass der Stein mit der Nummer \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 umgefallen ist. \par Diese Annahme nennt man die \plain\f4\fs28\cf0\b Induktionsvoraussetzung\plain\f4\fs28\cf0 . \par \par Unter dieser Voraussetzung m\'fcssen wir zeigen, dass auch der Stein mit der \par Nummer \plain\f3\fs28\cf1\i n + \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 umfallen wird, d.h. wir f\'fchren den sogenannten \plain\f4\fs28\cf0\b Induktions- \par schluss\plain\f4\fs28\cf0 durch. \par \par In unserer Situation bedeutet das: Wir nehmen an, die Summe der ersten \par \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 Quadratzahlen ist in der Tat durch \par 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\plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 ein. \par \par Der Beweis ist nun gar nicht: Wir d\'fcrfen annehmen, dass die Summe der \par Quadratzahlen von \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 bis \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 gegeben durch \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf4 {\pntext\f1\'b7\tab}Quadrate_bis_n \par \pard\ri4\plain\f4\fs28\cf0 \tab \tab \par Um die Summe der Quadratzahlen von \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 bis \plain\f3\fs28\cf1\i n + \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 zu berechnen, m\'fcssen wir \par also zu der Formel \plain\f5\fs28\cf4 Quadrate_bis_n \plain\f4\fs28\cf0 nur noch das Quadrat von \plain\f3\fs28\cf1\i n + \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 hinzu- \par addieren: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf4 {\pntext\f1\'b7\tab}Summe:= Quadrate_bis_n + (n + 1)^2 \par \pard\ri4\plain\f4\fs28\cf0 \par Nun pr\'fcfen wir 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\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf4 {\pntext\f1\'b7\tab}expand(Summe) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf4 {\pntext\f1\'b7\tab}expand(1/6 * (n + 1) * (n + 1 + 1) * (2 * (n + 1) + 1)) \par \pard\ri4\plain\f4\fs28\cf0 \par Die beiden Ausdr\'fccke sind gleich! Damit ist die Behauptung bewiesen. \par \par Nun zu einem weiteren Beispiel: Wir wollen beweisen, dass die Formel \par \pard\li2500\ri4\plain\f4\fs28\cf0 {\pict\wmetafile8\picw3740\pich1545\picscalex99\picscaley99\picwgoal2132\pichgoal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}\plain\f4\fs28\cf0 \tab \par \pard\ri4\plain\f4\fs28\cf0 korrekt ist, d.h. dass die Summe der Zahlen von \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 bis \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 gerade durch \par die rechte Seite der obigen Gleichung gegeben ist. \par \par Wir k\'fcrzen die Vorgehensweise jetzt st\'e4rker ab und gestalten den Beweis \par formaler: \par \par \plain\f4\fs28\cf0\b Induktionsanfang\plain\f4\fs28\cf0 (zeige die Behauptung f\'fcr \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 \plain\f3\fs28\cf1 =\plain\f4\fs28\cf0 \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 ): Die Summe der Zahlen \par von \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 bis \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 ist sicherlich \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 . Wir setzen den Wert \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 \plain\f3\fs28\cf1 =\plain\f4\fs28\cf0 \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 in die rechte Seite der \par obigen Gleichung ein und erhalten: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf4 {\pntext\f1\'b7\tab}Summe_1_bis_n:= 1/2 * n * (n + 1) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf4 {\pntext\f1\'b7\tab}subs(Summe_1_bis_n, n = 1) \par \pard\ri4\plain\f4\fs28\cf0 \par Damit ist der Induktionsanfang erledigt. \par \par \plain\f4\fs28\cf0\b Induktionsvoraussetzung:\plain\f4\fs28\cf0 Die Summe der Zahlen von 1 bis n ist \par \pard\li3000\ri4\plain\f4\fs28\cf0 {\pict\wmetafile8\picw2316\pich1279\picscalex99\picscaley99\picwgoal1316\pichgoal729 010009000003A903000008001C0000000000050000000B0200000000050000000C02FF040C0903 0000001E00050000000C0206051309050000000B0200000000030000001E00050000000C020D05 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}\plain\f4\fs28\cf0 \par \pard\ri4\plain\f4\fs28\cf0 \par \plain\f4\fs28\cf0\b Induktionsschluss\plain\f4\fs28\cf0 : Setzen wir in die Formel f\'fcr \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 den Wert \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 \plain\f3\fs28\cf1\i + \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 ein, so \par erhalten wir: \tab \tab \tab \tab \tab \par \pard\li2500\ri4\plain\f4\fs28\cf0 {\pict\wmetafile8\picw4156\pich1279\picscalex99\picscaley99\picwgoal2361\pichgoal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}\plain\f4\fs28\cf2 ::::::::::::::::::\plain\f4\fs28\cf1 ( * ) \par \pard\ri4\plain\f4\fs28\cf0 \tab \tab \par Unter Verwendung der Induktionsvoraussetzung ergibt sich die Summe der \par Zahlen von \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 bis \plain\f3\fs28\cf1\i n\plain\f4\fs28\cf0 \plain\f3\fs28\cf1\i + \plain\f3\fs28\cf1 1\plain\f4\fs28\cf0 zu: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf4 {\pntext\f1\'b7\tab}Summe:= Summe_1_bis_n + (n + 1) \par \pard\ri4\plain\f4\fs28\cf0 \par Wir m\'fcssen zeigen, dass dieser Ausdruck und \plain\f4\fs28\cf1 ( * )\plain\f4\fs28\cf0 \'fcbereinstimmen: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf4 {\pntext\f1\'b7\tab}expand(Summe) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf4 {\pntext\f1\'b7\tab}expand(1/2 * (n + 1) * (n + 1 + 1)) \par \pard\ri4\plain\f4\fs28\cf0 \par was offensichtlich der Fall ist. Damit ist die Behauptung bewiesen. \par \par \plain\f4\fs28\cf0\b Aufgabe:\plain\f4\fs28\cf0 \tab Beweise analog zu der obigen Vorgehensweise die Formel \par \par \pard\li2500\ri4\plain\f4\fs28\cf0 {\pict\wmetafile8\picw4268\pich1609\picscalex99\picscaley99\picwgoal2428\pichgoal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}\plain\f4\fs28\cf0 \par \pard\ri4\plain\f4\fs28\cf0 \par \plain\f5\fs20\cf0\b ________________________________________________________________________________ \par \plain\f4\fs22\cf0 \par \plain\f4\fs22\cf5\b Anmerkungen:\plain\f4\fs22\cf5 \par \plain\f4\fs20\cf5 \par \plain\f4\fs20\cf5\b 1\plain\f4\fs20\cf5 . Weitere Anregungen finden Sie in der Buchreihe \plain\f4\fs20\cf4 Mathematik 1 x anders\plain\f4\fs20\cf5 . In dieser Reihe \par wird eine Vielzahl unterschiedlichster mathematischer Probleme mit MuPAD gel\'f6st. Die \par B\'fccher k\'f6nnen unter \plain\f6\fs20\cf1 www.schule.mupad.de/literatur\plain\f4\fs20\cf5 kostenfrei kopiert werden. \par \par \plain\f4\fs20\cf5\b 2\plain\f4\fs20\cf5 . Viele weitere praxisorientierte Aufgaben finden sich unter der Web-Adresse \par \plain\f4\fs20\cf5\b \par \pard\li2500\ri4\plain\f4\fs20\cf5\b http://www.learn-line.nrw.de\plain\f5\fs28\cf4 \par \pard\ri4\plain\f5\fs20\cf0\b ________________________________________________________________________________ \par \plain\f5\fs28\cf4 \par }