Page 52 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

And in fact, if we keep pairing the numbers this way, each pair will always have the same sum of 101.

How many pairs are there? Since there are 100 numbers total, there are exactly 50 pairs. So, the sum
of all the pairs is equal to the sum of each pair times the total number of pairs,

101 ∗ 50 = 5050.

In fact, we can always play the game of addition this way to compute that for any n ∈ N , the sum of
all integers from 1 to n is

n(n +1)
.
2

We will prove this formula using induction.

A proof by induction is like riding a mathemagical escalator.

1. First, we must step onto the escalator; we do this by proving the base case. The base case

is the first positive integer, so we must prove that the statement is true when n =1.
Sometimes, your statement starts at a later integer, like n =2 or n =10, 000; it could also
start at a negative integer like n = −1. That’s okay, you are just jumping onto the escalator
at a different height. This starting point will be your base case.

52