With two attendees, there is only one handshake. When a third attendee arrives, the new guest must shake hands with two people, so there is a total of three handshakes. When the fourth guest arrives, he or she must shake hands with each of the three guests already in the room, and when the ith person enters the room, that guest must shake hands with all the i – 1 guests previously in the room. So with n + 1 people at the party, the number of handshakes equals the sum 1 + 2 + …+ n = Σk and, as noted before, is expressed as a function of n by Σk = n(n + 1)/2.