If G be the G.M. between two given numbers and A1, A2 be the two A.Ms between them,prove that
G2(2A1-A2) (2A-A2)
To prove the expression G²(2A₁ - A₂)(2A - A₂), we need to manipulate the given information about the geometric mean (G) and two arithmetic means (A₁ and A₂).
Let's start by determining the relationship between G, A₁, and A₂. The geometric mean (G) between two numbers can be expressed as the square root of their product. Therefore, we can write it as:
G = √(A₁ * A₂)
Next, let's express A₁ and A₂ in terms of G. Since A₁ and A₂ are the two arithmetic means between the numbers, we can write:
A₁ = (G + G)/2 = 2G/2 = G
A₂ = (G + 2G)/2 = 3G/2
Now, we can substitute these values of A₁ and A₂ back into the original expression:
G²(2A₁ - A₂)(2A - A₂)
= G²(2G - 3G/2)(2A - 3G/2)
= G²(G/2)(2A - 3G/2)
= G²G(2A - 3G/2)
= G³(2A - 3G/2)
Finally, we can simplify the expression:
G³(2A - 3G/2)
= (G³ * 2A) - (G³ * 3G/2)
= 2G³A - (3/2)G⁴
Thus, we have proved the expression G²(2A₁ - A₂)(2A - A₂) is equivalent to 2G³A - (3/2)G⁴.
Note: The proof involves expressing the arithmetic means (A₁ and A₂) in terms of the geometric mean (G) and substituting these values back into the original expression.