Q. 293.8( 64 Votes )

# If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are .

Answer :

Given: *A* and *G* are A.M. and G.M. between two positive numbers.

Let the two numbers be *a* and *b*.

∴ AM = A = —1

GM = G = √ab —2

From eq-1 and eq-2, we get

*a* + *b* = 2*A* —3

*ab* = *G*^{2} —4

Substituting the value of *a* and *b* from eq-3 and eq-4 in

(*a* – *b*)^{2} = (*a* + *b*)^{2} – 4*ab*, we get

(*a* – *b*)^{2} = 4*A*^{2} – 4*G*^{2} = 4 (*A*^{2}–*G*^{2})

(*a* – *b*)^{2} = 4 (*A* + *G*) (*A* – *G*)

(a – b) = 2 —5

From eq-3 and eq-5, we get

2a = 2A + 2

⇒ a = A+2

Substituting the value of *a* in eq-3, we get

b = 2A – A - = A –

Thus, the two numbers are .

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