Problema Solution
If 'A' and 'G' are Arithmetic mean and Geometric mean between two numbers then prove that the numbers are A+or-sqrt of (A+G)(A-G)
Answer provided by our tutors
Let the two numbers be x and y.
A = (x + y)/2
=> x + y = 2A
=> x = 2A - y
G = √xy
So, xy = G2
x = G2 / y
Now, 2A - y = G2 / y
2Ay - y2 = G2
y2 - 2Ay + G2 = 0
y = [2A ± √{(2A)2 - 4(1)(G2)}] / 2(1)
y = [2A ± 2√(A2 - G2)] / 2
y = A ± √(A2 - G2)
When y = A + √(A2 - G2), then x = 2A - y = A - √(A2 - G2)
When y = A - √(A2 - G2), then x = 2A - y = A + √(A2 - G2)
So, the two numbers are A ± √(A2 - G2)
A ± √(A2 - G2)
= A ± √{(A+G)(A-G)}