The Most Important Sum

1+2+3+…+(n-1)+n = n(n+1)2

Applications:

• n people; how many handshakes?• (equivalent: n cities to directly connect with

phone cables; how many cables needed?)• Arranging n cards in order: how many compares?• Appending to a list of names: how many steps?• Keeping students busy (anecdote: Gauss)

Three derivations

• Gauss' approach: 1 + 2 + 3 + … + (n-2) + (n-1) + n = S

n +(n-1)+(n-2)+ … + 3 + 2 + 1 = S

(n+1)+(n+1)+(n+1)+…+(n+1) + (n+1) + (n+1) = 2S

n(n+1) = S 2

Three Derivations (part 2)

.

.

.. . .

.

.

.

1 2 3 n-1 n

123...n-1n

Three Derivations (part 2)

.

.

.. . .

.

.

.

1 2 3 n-1 n

123...n-1n

n(n+1) = S 2

.

.

.

.

.

.

. . .

Three Derivations (part 3)

Count the ways to create a handshake:– Choose the 1st person (n choices)– Choose the 2nd person (n-1 remaining

choices(That's n(n-1) ways to choose one then the other.)

– But this overcounts: "Choose Alice then Bob" and "Choose Bob then Alice" are the same handshake; divide by 2 to correct.

S = n(n+1) 2