To find two harmonic means between two numbers, you need to create a harmonic progression (HP) with those two numbers as the first and last terms, and the two harmonic means as the intermediate terms.
Here's a breakdown of the process:
Steps to Find Two Harmonic Means
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Understand Harmonic Progression (HP): A sequence of numbers is in HP if the reciprocals of the terms are in Arithmetic Progression (AP).
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Set up the HP: Let 'a' and 'b' be the two numbers between which you want to insert two harmonic means, H1 and H2. The HP will be: a, H1, H2, b
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Convert to AP: Take the reciprocals of the HP to form an AP: 1/a, 1/H1, 1/H2, 1/b
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Determine the Common Difference (d): In this AP, 1/a is the first term, and 1/b is the fourth term. Use the AP formula:
Termn = a + (n-1)d
Where:
- Termn is the nth term of the AP (in this case, 1/b)
- a is the first term of the AP (in this case, 1/a)
- n is the number of terms (in this case, 4)
- d is the common difference (what we want to find)
Therefore: 1/b = 1/a + (4-1)d => 1/b = 1/a + 3d
Solving for d: 3d = 1/b - 1/a => 3d = (a - b) / ab => d = (a - b) / 3ab
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Find 1/H1 and 1/H2: Now that you know the common difference, you can find the intermediate terms of the AP.
- 1/H1 = 1/a + d = 1/a + (a - b) / 3ab = (3b + a - b) / 3ab = (a + 2b) / 3ab
- 1/H2 = 1/a + 2d = 1/a + 2(a - b) / 3ab = (3b + 2a - 2b) / 3ab = (2a + b) / 3ab
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Find H1 and H2: Take the reciprocals of 1/H1 and 1/H2 to find the harmonic means.
- H1 = 3ab / (a + 2b)
- H2 = 3ab / (2a + b)
Example:
Let's find two harmonic means between 2 and 6.
- a = 2, b = 6
- d = (2 - 6) / (3 2 6) = -4 / 36 = -1/9
- 1/H1 = 1/2 + (-1/9) = (9 - 2) / 18 = 7/18 => H1 = 18/7
- 1/H2 = 1/2 + 2*(-1/9) = (9 - 4) / 18 = 5/18 => H2 = 18/5
Therefore, the two harmonic means between 2 and 6 are 18/7 and 18/5.
