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Suppose you have people in a room and each person shakes hands with each other person once. How many handshakes do you get in total? The first person shakes hands with other people, the second shakes hands with the remaining people, the second shakes hands with remaining people, etc, giving a total of
But we can also look at this in another way: each person shakes hands with others and there are people, giving handshakes. But this counts every handshake twice, so we need to divide by 2, giving a total of
Putting these two arguments together, we have just come up with the formula for summing the first integers and we’ve proved that it is correct:
Maths can be so easy!