Final answer:
To find the least positive integer b greater than 40 that makes the Heronian mean H(40, b) an integer, we use the formula H(a, b) = (a + sqrt(ab) + b) / 3, substitute a with 40, and solve for b by testing consecutive integers starting from 41.
Step-by-step explanation:
The question asks to find the least positive integer b greater than 40 such that when plugged into the Heronian mean formula with a being 40 results in an integer value. The Heronian mean is defined as H(a, b) = (a + sqrt(ab) + b) / 3.
To solve for b, we first substitute a with 40 in the Heronian mean formula and set the expression equal to an integer:
H(40, b) = (40 + sqrt(40*b) + b) / 3 = integer
Multiplying both sides of the equation by 3, and subtracting 40 and b from both sides, we get:
3*integer - 40 - b = sqrt(40*b)
Squaring both sides, we obtain an equation in terms of b:
(3*integer - 40 - b)² = 40*b
To find the least b, we need to test consecutive integers for the value on the right side, starting from 41 onwards, until the equation is satisfied with b as an integer.
Once the correct value for b is found, it will be the smallest integer greater than 40 for which the Heronian mean with a=40 is also an integer.