Additionstabelle/N/Volle Summe/Aufgabe

Wir betrachten die naürliche Additionstabelle bis zu einer bestimmten Zahl ${}n$, also

${}+$	${}1$	${}2$	${}3$	${}\ldots$	${}n-1$	${}n$
${}1$	${}1+1$	${}1+2$	${}1+3$	${}\ldots$	${}1+n-1$	${}1+n$
${}2$	${}2+1$	${}2+2$	${}2+3$	${}\ldots$	${}2+n-1$	${}2+n$
${}3$	${}3+1$	${}3+2$	${}3+3$	${}\ldots$	${}3+n-1$	${}3+n$
${}\ldots$	${}\vdots$	${}\vdots$	${}\vdots$	${}\vdots$	${}\vdots$	${}\vdots$
${}n-1$	${}n-1+1$	${}n-1+2$	${}n-1+3$	${}\ldots$	${}n-1+n-1$	${}n-1+n$
${}n$	${}n+1$	${}n+2$	${}n+3$	${}\ldots$	${}n+n-1$	${}n+n$

Zeige durch Induktion, dass die Gesamtsumme aller in der Tabelle auftretenden Summen gleich ${}(n+1)n^{2}$ ist, also

${}\sum _{1\leq i\leq n,\,1\leq j\leq n}(i+j)=(n+1)n^{2}\,.$

Zur Lösung, Alternative Lösung erstellen