Question

What is the average (arithmetic mean) of eleven consecutive integers? (1) the average of the first nine integers is 7. (2) the average of the last nine integers is 9?

Answer 1

Since eleven is odd, there must be a centra element, which in this case is the sixth. So, we can write 11 consecutive integers as

$x-5, x-4, x-3, x-2, x-1, x, x+1, x+2, x+3, x+4, x+5$

Since the arithmetic mean is the sum of the values, divided by how many they are, we have

$M = \dfrac{x-5, x-4, x-3, x-2, x-1, x, x+1, x+2, x+3, x+4, x+5}{11} = \dfrac{11x}{11} = x$

So, the mean of 11 consecutive integers is the 6th integer.