Question

17.Arithmetic Mean and Geometric Mean of two positive real numbers a and b are equal if and only if *

1.a=b
2.a<b
3.a>b
4=None of these

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Answer 1

Answer:

a = b

Step-by-step explanation:

Arithmetic mean = (a + b)/2

Geometric mean = √ab

       Since both are equal:

⇒ (a + b)/2  = √ab

⇒ a + b = 2√ab

⇒ a + b - 2√ab = 0

⇒ (√a - √b)² = 0

⇒ √a - √b = 0

⇒ √a = √b

⇒  a = b

     

Answer 2

Answer:

1.a=b

Step-by-step explanation:

Airthmetic mean = Geometric mean

(a+b)/2 = √ab

(a+b) = 2√ab

square of both sides,

(a+b)^2 = (2√ab)^2

a^2+b^2+2ab = 4ab

a^2+b^2+2ab-4ab= 0

a^2+b^2-2ab= 0

(a-b)^2 = 0

(a-b) = 0

a = b