What is the average of all the integer's of 13 to 37?

Answer 1

To find the (mean) average, add all the numbers and divide by the number of numbers.

The sum of a series of digits (in arithmetic progression, like 13, 14, 15, ... 37) is

sum = (first + last) x number_of_digits / 2

So their average is:

average = ((first + last) x number_of_digits / 2) / number_of_digits

= (first + last) / 2

= average of first and last digits!

So the average of the numbers 13, 14, 15, ..., 37 is:

average = (13 + 37) / 2

= 50 / 2

= 25

To find the sum of n digits starting with m:

Sum = m + (m+1) + ... + (m+n-2) + (m+n-1)

Rewrite the sum in reverse order:

Sum2 = Sum = (m+n-1) + (m+n-2) + ... + (m+1) + m

Add the two sums, term by term:

Sum + Sum2 = 2 Sum = (m + (m+n-1)) + (m + (m+n-1)) + ... + (m + (m+n-1)) + (m + (m+n-1))

There are n terms, all (m + (m+n-1)), so:

2 Sum = (m + m+n-1) x n

Sum = (m + m+n-1) x n / 2

But n is the number of digits, m is the first number and (m+n-1) is the last, so:

Sum = (first + last) x number_of_digits / 2