Simple average question

[mchow]

Hi,
if a question says:
If the average of the first 16 positive integers is subtracted from the average next 16 positive integers, what is the result?

How come the average of each series is computed as:
(1 + 16)/2 and (17 + 32)/2 not (1+2+3...+16)/16 and (17+18+...+32)/16?

Thanks!

[Feep]

Because they're the same thing.

For a better example, let's look at the sequence 3, 4, 5, 6, 7. The average here is, clearly, 5. But notice that the elements are evenly spaced around the average in pairs: 3 is two below, 7 is two above. 4 is one below, 6 is one above. They cancel each other out. 1, 2, 3, 4, 5, 6, 7, 8, 9 is no different. Corresponding pairs equally spaced from the center do not affect the average.

So, the quicker computation uses this shortcut. Because all the numbers are equally spaced, we may assume that each pair will have the same average. 1 and 16 has the same average as 2 and 15, as 3 and 14, as 4 and 13, and so on. It's just a quicker method.