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What is the sum of all odd numbers between 0 and 100?

1 Answer
Sep 17, 2015

First, notice an interesting pattern here:

#1, 4, 9, 16, 25, ...#

The differences between perfect squares (starting at #1-0 = 1#) is:

#1, 3, 5, 7, 9, ...#

The sum of #1+3+5+7+9# is #25#, the #5^"th"# nonzero square.

Let's take another example. You can quickly prove that:

#1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100#

There are #(19+1)/2 = 10# odd numbers here, and the sum is #10^2#.

Therefore, the sum of #1 + 3 + 5 + ... + 99# is simply:

#((99+1)/2)^2 = color(blue)(2500)#

Formally, you can write this as:

#color(green)(sum_(n=1)^N (2n-1) = 1 + 3 + 5 + ... + (2N - 1) = ((N+1)/2)^2)#

where #N# is the last number in the sequence and #n# is the index of each number in the sequence. So, the #50^"th"# number in the sequence is #2*50 - 1 = 99#, and the sum all the way up to that is #((99 + 1)/2)^2 = 2500#.