Mean Value Theorem

To solve the problem, we need to find the value of G⁴₁+G⁴₂+G⁴₃+G²₁G²₃ where G₁, G₂, G₃ are the geometric means of two distinct positive numbers A and B.

1. Define the Numbers:

Let the two distinct positive numbers be A and B.

2. Find the Geometric Means:

The geometric means G₁, G₂, G₃ can be defined as follows:

• G₁=A^1/4B^3/4
• G₂=A^1/2B^1/2
• G₃=A^3/4B^1/4

3. Calculate G⁴₁, G⁴₂, and G⁴₃:

• G⁴₁=(A^1/4B^3/4)⁴=A¹B³=AB³
• G⁴₂=(A^1/2B^1/2)⁴=A²B²
• G⁴₃=(A^3/4B^1/4)⁴=A³B¹=A³B

4. Sum the Fourth Powers:

Now we sum these results: G⁴₁+G⁴₂+G⁴₃=AB³+A²B²+A³B

5. Calculate G²₁G²₃:

Now, we need to calculate G²₁G²₃:

• G²₁=(A^1/4B^3/4)²=A^1/2B^3/2
• G²₃=(A^3/4B^1/4)²=A^3/2B^1/2
• Therefore, G²₁G²₃=(A^1/2B^3/2)(A^3/2B^1/2)=A^1/2+3/2B^3/2+1/2=A²B²

6. Combine All Terms:

Now we combine all the terms: G⁴₁+G⁴₂+G⁴₃+G²₁G²₃=(AB³+A²B²+A³B)+A²B²

This simplifies to: AB³+2A²B²+A³B

7. Factor the Expression:

We can factor this expression: =AB(A²+2AB+B²)=AB(A+B)²

Final Answer:

Thus, the expression G⁴₁+G⁴₂+G⁴₃+G²₁G²₃ is equal to: AB(A+B)²