A Graph Theory Interpretation of the Sum of the First n Integers

n

1

0

+

1

=

1

=



2



A complete graph on

n+1

vertices has



n+1

2



edges. This number is also the sum of the first

n

integers, as shown here: the complete graphs are built up by adding one vertex at a time, and the colors show that the

th

(n+1)

vertex adds

n

edges, thus showing visually that

n

∑

i=1

i=

n+1

2



.