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)

u rich

step-by-step explanation: