The Airthmetic mean between two

positive numbers a and b

is to twice their Geometric mean.

prove that a:b = (2+3) : (2-13)

Step-by-step explanation:

The arithmetic mean is (a + b) / 2.

The geometric mean is √ (ab).

If the arithmetic mean is twice the geometric mean:

(a + b) / 2 = 2√ (ab)

a + b = 4√ (ab)

(a + b) ² = 16ab

a² + 2ab + b² = 16ab

a² - 14ab + b² = 0

Complete the square:

a² - 14ab + 49b² + b² = 49b²

(a - 7b) ² = 48b²

a - 7b = √48 b

a = (7 + √48) b

a/b = 7 + √48

a/b = 7 + 4√3

We can show this equals (2+√3) / (2-√3) by multiplying by (2-√3) / (2-√3).

a/b = (7 + 4√3) * (2 - √3) / (2 - √3)

a/b = (14 - 7√3 + 8√3 - 12) / (2 - √3)

a/b = (2 + √3) / (2 - √3)