Question

If A and G is AM and GM of two positive real numbers then A and G are related as ​

Answer 1

Answer:

Given A and G are AM and GM between two numbers. Let the two number be a and b

then A=

2

a+b

G=

ab

⇒a+b=2A(1)

⇒ab=G

2

(2)

⇒(a−b)

2

=(a+b)

2

−4ab

⇒(a−b)

2

=(2A)

2

−4G

2

⇒(a−b)

2

=4(A

2

−G

2

)

⇒(a−b)=±2

A

2

−G

2

(3)

Taking (1) and (3)

⇒a−b=2

A

2

−G

2

⇒a+b=2A

Adding both 2a=2A+2

A

2

−G

2

⇒a=A+

A

2

−G

2

Putting this value of a in equation (1)

⇒a+b=2A

⇒A+

A

2

−G

2

+b=2A

⇒b=A−

A

2

−G

2

= A−

(A+G)(A−G)

.

Hence, the numbers are A±

A

2

−G

2

Step-by-step explanation:

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