Arithmetic means are the numbers that would be part of an arithmetic sequence where 19 is the 1st term and 52 is the 5th term (in this case) - notice that gives tree numbers in the sequence between the two. Like this:
So:
19 + x + x + x + x = 52
19 + 4x = 52
4x = 52 - 19
4x = 33
x = 33/4 = 8.25
So the 1st arithmetic mean is 19 + 8.25 = 27.25
The 2nd arithmetic mean is 27.25 + 8.25 = 35.5
The 3rd arithmetic mean is 35.5 + 8.25 = 43.75
And then we double check things by adding 43.75 + 8.25 and we do indeed get 52
