Sum of the first n numbers/Doubling/Exercise

Prove the formula

${}\sum _{k=1}^{n}k={\frac {n(n+1)}{2}}\,$

without induction, by considering the following table

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

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