Question:

Let A₁ and A₂ be two arithmetic means and G1, G2, G3 be three geometric means of two distinct positive numbers. Then $ G_{1}^{4} + G_{2}^{4}+ G_{3}^{4} +G_{1}^{2}G_{3}^{2} $ is equal to

Updated On: Mar 21, 2025
  • 2(A1 + A2) G1G3
  • (A1 + A2) $ G_{1}^{2}G_{3}^{2} $
  • (A1 + A2)2 G1G3
  • 2(A1 + A2) $ G_{1}^{2}G_{3}^{2} $
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The Correct Option is C

Solution and Explanation

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=A1/4B3/4
  • G2=A1/2B1/2
  • G3=A3/4B1/4

3. Calculate G41, G42, and G43:

  • G41=(A1/4B3/4)4=A1B3=AB3
  • G42=(A1/2B1/2)4=A2B2
  • G43=(A3/4B1/4)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=(A1/4B3/4)2=A1/2B3/2
  • G23=(A3/4B1/4)2=A3/2B1/2
  • Therefore, G21G23=(A1/2B3/2)(A3/2B1/2)=A1/2+3/2B3/2+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

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