Question

What is the arithmetic mean of the first million odd numbers

Answer 1

It's 1,000,000: 

the sum is 1 + 3 + 5 + ... 1999999 

= 1 + 2 + ... + 2000000 - 2*(1 + ... + 1000000) 

= 2000000*2000001/2 - 2*(1000000*1000001 / 2) 

= 1000000*(2000001 - 1000001) = 1000000000000 (one million million) 

and dividing this by one million to get the average, 

you get one million