find two numbers whose A.M exceeds their G.M by 1/2 & their H.M by 25/26.
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find two numbers whose A.M exceeds their G.M by 1/2 & their H.M by 25/26.
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Let's find two numbers, A and B, such that their Arithmetic Mean (A.M) exceeds their Geometric Mean (G.M) by 1/2, and their Harmonic Mean (H.M) exceeds by 25/26.
The Arithmetic Mean (A.M) of two numbers A and B is (A + B) / 2.
The Geometric Mean (G.M) of two numbers A and B is the square root of (A * B), which is √(A * B).
The Harmonic Mean (H.M) of two numbers A and B is 2 / [(1/A) + (1/B)].
We are given two conditions:
1. (A + B) / 2 - √(A * B) = 1/2
2. 2 / [(1/A) + (1/B)] + 25/26
We can rewrite the second condition as:
2 / [(1/A) + (1/B)] = 25/26
Now, let's solve these equations simultaneously:
From the first equation:
(A + B) / 2 - √(A * B) = 1/2
Multiply both sides by 2 to eliminate the fraction:
A + B - 2√(A * B) = 1
Now, let's square both sides of this equation to get rid of the square root:
(A + B - 2√(A * B))^2 = 1^2
A^2 + B^2 + (2√(A * B))^2 + 2 * A * B - 2 * A * B - 4√(A * B) * (A + B) = 1
A^2 + B^2 + 4 * A * B - 4√(A * B) * (A + B) = 1
Now, from the second equation:
2 / [(1/A) + (1/B)] = 25/26
Cross-multiply:
2 * 26 = 25 * [(1/A) + (1/B)]
52 = 25 * [(1/A) + (1/B)]
Now, we can solve this equation for (1/A) + (1/B):
(1/A) + (1/B) = 52 / (25 * 25)
(1/A) + (1/B) = 52 / 625
Now, let's solve these two equations simultaneously to find A and B. We have a system of equations:
1. A^2 + B^2 + 4 * A * B - 4√(A * B) * (A + B) = 1
2. (1/A) + (1/B) = 52 / 625
Solving this system of equations may require numerical methods or a calculator, and the solutions may not be simple integers.
[tex]\tiny\tt\red{Hope \: it \: helps}[/tex]
[tex]ᏗᏁᏕᏇᏋᏒ[/tex]
Let's find two numbers, A and B, such that their Arithmetic Mean (A.M) exceeds their Geometric Mean (G.M) by 1/2, and their Harmonic Mean (H.M) exceeds by 25/26.
The Arithmetic Mean (A.M) of two numbers A and B is (A + B) / 2.
The Geometric Mean (G.M) of two numbers A and B is the square root of (A * B), which is √(A * B).
The Harmonic Mean (H.M) of two numbers A and B is 2 / [(1/A) + (1/B)].
We are given two conditions:
1. (A + B) / 2 - √(A * B) = 1/2
2. 2 / [(1/A) + (1/B)] + 25/26
We can rewrite the second condition as:
2 / [(1/A) + (1/B)] = 25/26
Now, let's solve these equations simultaneously:
From the first equation:
(A + B) / 2 - √(A * B) = 1/2
Multiply both sides by 2 to eliminate the fraction:
A + B - 2√(A * B) = 1
Now, let's square both sides of this equation to get rid of the square root:
(A + B - 2√(A * B))^2 = 1^2
A^2 + B^2 + (2√(A * B))^2 + 2 * A * B - 2 * A * B - 4√(A * B) * (A + B) = 1
A^2 + B^2 + 4 * A * B - 4√(A * B) * (A + B) = 1
Now, from the second equation:
2 / [(1/A) + (1/B)] = 25/26
Cross-multiply:
2 * 26 = 25 * [(1/A) + (1/B)]
52 = 25 * [(1/A) + (1/B)]
Now, we can solve this equation for (1/A) + (1/B):
(1/A) + (1/B) = 52 / (25 * 25)
(1/A) + (1/B) = 52 / 625
Now, let's solve these two equations simultaneously to find A and B. We have a system of equations:
1. A^2 + B^2 + 4 * A * B - 4√(A * B) * (A + B) = 1
2. (1/A) + (1/B) = 52 / 625
Solving this system of equations may require numerical methods or a calculator, and the solutions may not be simple integers.....