Question

22. What is the average(arithmetic mean) of all the integers
from -39 to 40, inclusive?

Answer 1

This range contains 80 numbers (40 in 1-40, and another 40 in -39-0).

We have

S = 1 + 2 + 3 + … + 38 + 39 + 40

S = 40 + 39 + 38 + … + 3 + 2 + 1

==> 2S = (1 + 40) + (2 + 39) + … + (39 + 2) + (40 + 1)

==> 2S = 40 × 41

==> S = 820

and

T = 0 + 1 + 2 + … + 37 + 38 + 39

T = 39 + 38 + 37 + … + 3 + 2 + 1

==> 2T = (0 + 39) + (1 + 38) + … + (38 + 2) + (39 + 0)

==> 2T = 40 × 39

==> T = 780

The average of the numbers in this range is equal to the total divided by the amount of numbers in the range, so

average = (40 + 39 + … + 1 + 0 + (-1) + … + (-38) + (-39))/80

average = (S - T )/80

average = (820 - 780)/80

average = 40/80 = 1/2