Question

Q.if A and G be A.M. and G.M., respectively between two positive
numbers,prove that the number are A plusminus underroot(A+G)(A-G) .

Answer 1

Let the 2 positive numbers be a and b .

A is the arithmetic mean .

Arithmetic mean = ( sum of numbers ) / ( number of terms )

A = ( a + b )/2

G is the geometric mean .

Geometric mean = n th root of ( product of numbers )

G = √ab

A ± √( A + G )( A - G )

A ± √( A² - G² )

⇒ ( a + b )/2 ± √( a + b )²/4 - ab )

⇒ ( a + b )/2 ± √( a + b )² - 4 ab )/√4

⇒ ( a + b )/2 ± √( a² + b² -  2 ab )/2

⇒ ( a + b )/2 ± √( a - b )²/2

⇒ ( a + b )/2 ± ( a - b )/2

Either :

( a + b )/2 + ( a - b )/2

= ( 2 a )/2

= a

Or:

( a + b )/2 - ( a - b)/2

⇒ ( b + b )/2

⇒ 2b/2

⇒ b

Hence a , b = A ± √( A + G )( A - G )

Answer 2

ANSWER:-----------

A = ( a + b )/2 G is the geometric mean .

G = √abA ± √( A + G )( A - G )A ± √( A² - G² )

⇒ ( a + b )/2 ± √( a + b )²/4 - ab )

⇒ ( a + b )/2 ± √( a + b )² - 4 ab )/√4

⇒ ( a + b )/2 ± √( a² + b² -  2 ab )/2

⇒ ( a + b )/2 ± √( a - b )²/2

⇒ ( a + b )/2 ± ( a - b )/2

hope it helps:-------

T!—!ANKS!!!