Math CentralQuandaries & Queries


Question from Faizan, a student:

What is the average of first 100 even numbers?

The way to proceed by definition is to add 2+4+6+… + 198 + 200 and divide by 100. Perhaps this is an exercise in using the standard technique for adding the terms of an arithmetic sequence (which can be found in textbooks or on the web). But an easier way to find averages is to compute the average of averages. Here I would find the average of 2 and 200, then 4 and 198, then 6 and 196, …. Then it is easy to take the average of those 50 averages.



The "easy" conceptual answer is to use symmetry - if you take 2 (as opposed to 0) as the first even number, then 100 is the fiftieth, 102 the 51st, and they can be arranged into pairs each of which averages 101. Also the fact that any weighted mean of equal numbers is the same number.

However, this method, while "obvious" to the seasoned mathematician requires quite a lot of things to be checked! You're better off using the following three "tools," all of which I assume you know.

(1) The definition of the mean: it's the sum of the N numbers divided by the count, N.

(2) The formula for the sum of the first N whole numbers , N(N+1)/2

(3) The distributive law: (2+4+6+... ) = (2*1 + 2*2 + 2*3 + ...) = 2(1+2+3+...)

Good Hunting!

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