Answer :
To solve for the two numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] given the harmonic mean (HM), arithmetic mean (AM), and geometric mean (GM), we'll utilize their definitions and the given condition.
Step 1: Recall the definitions
1. Harmonic Mean (HM) between [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ \text{HM} = \frac{2ab}{a + b} \][/tex]
Given [tex]\(\text{HM} = \frac{16}{5}\)[/tex].
2. Arithmetic Mean (AM) between [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ \text{AM} = \frac{a + b}{2} \][/tex]
Denote this as [tex]\( A \)[/tex].
3. Geometric Mean (GM) between [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ \text{GM} = \sqrt{ab} \][/tex]
Denote this as [tex]\( G \)[/tex].
4. Given condition:
[tex]\[ 2A + G^2 = 26 \][/tex]
Step 2: Set up equations based on the given information
1. From the HM definition:
[tex]\[ \frac{2ab}{a + b} = \frac{16}{5} \][/tex]
Multiplying both sides by [tex]\( a + b \)[/tex]:
[tex]\[ 2ab = \frac{16}{5}(a + b) \][/tex]
Simplify it as:
[tex]\[ 10ab = 16(a + b) \][/tex]
[tex]\[ 5ab = 8(a + b) \tag{1} \][/tex]
2. From the AM definition, denote [tex]\( A \)[/tex] as:
[tex]\[ A = \frac{a + b}{2} \][/tex]
3. From the GM definition, denote [tex]\( G \)[/tex] as:
[tex]\[ G = \sqrt{ab} \][/tex]
4. The given condition:
[tex]\[ 2A + G^2 = 26 \][/tex]
Substitute [tex]\( A \)[/tex] and [tex]\( G \)[/tex]:
[tex]\[ 2\left(\frac{a + b}{2}\right) + (\sqrt{ab})^2= 26 \][/tex]
Simplify:
[tex]\[ a + b + ab = 26 \tag{2} \][/tex]
Step 3: Solve the equations
We now have two key equations:
[tex]\[ 5ab = 8(a + b) \tag{1} \][/tex]
[tex]\[ a + b + ab = 26 \tag{2} \][/tex]
1. Let's express [tex]\( b \)[/tex] in terms of [tex]\( a \)[/tex]:
From equation (2):
[tex]\[ b = \frac{26 - a - ab}{a} \][/tex]
2. Substitute into equation (1):
[tex]\[ 5a\left(\frac{26 - a - ab}{a}\right) = 8(a + \frac{26 - a - ab}{a}) \][/tex]
Simplify and solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- After solving these, the solutions are [tex]\( a = 2 \)[/tex] and [tex]\( b = 8 \)[/tex].
Final Solution: The numbers are 2 and 8.
Thus, given the conditions, the numbers that satisfy the given harmonic mean, arithmetic mean, geometric mean, and the equation [tex]\( 2A + G^2 = 26 \)[/tex] are:
[tex]\[ \boxed{2 \text{ and } 8} \][/tex]
Step 1: Recall the definitions
1. Harmonic Mean (HM) between [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ \text{HM} = \frac{2ab}{a + b} \][/tex]
Given [tex]\(\text{HM} = \frac{16}{5}\)[/tex].
2. Arithmetic Mean (AM) between [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ \text{AM} = \frac{a + b}{2} \][/tex]
Denote this as [tex]\( A \)[/tex].
3. Geometric Mean (GM) between [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ \text{GM} = \sqrt{ab} \][/tex]
Denote this as [tex]\( G \)[/tex].
4. Given condition:
[tex]\[ 2A + G^2 = 26 \][/tex]
Step 2: Set up equations based on the given information
1. From the HM definition:
[tex]\[ \frac{2ab}{a + b} = \frac{16}{5} \][/tex]
Multiplying both sides by [tex]\( a + b \)[/tex]:
[tex]\[ 2ab = \frac{16}{5}(a + b) \][/tex]
Simplify it as:
[tex]\[ 10ab = 16(a + b) \][/tex]
[tex]\[ 5ab = 8(a + b) \tag{1} \][/tex]
2. From the AM definition, denote [tex]\( A \)[/tex] as:
[tex]\[ A = \frac{a + b}{2} \][/tex]
3. From the GM definition, denote [tex]\( G \)[/tex] as:
[tex]\[ G = \sqrt{ab} \][/tex]
4. The given condition:
[tex]\[ 2A + G^2 = 26 \][/tex]
Substitute [tex]\( A \)[/tex] and [tex]\( G \)[/tex]:
[tex]\[ 2\left(\frac{a + b}{2}\right) + (\sqrt{ab})^2= 26 \][/tex]
Simplify:
[tex]\[ a + b + ab = 26 \tag{2} \][/tex]
Step 3: Solve the equations
We now have two key equations:
[tex]\[ 5ab = 8(a + b) \tag{1} \][/tex]
[tex]\[ a + b + ab = 26 \tag{2} \][/tex]
1. Let's express [tex]\( b \)[/tex] in terms of [tex]\( a \)[/tex]:
From equation (2):
[tex]\[ b = \frac{26 - a - ab}{a} \][/tex]
2. Substitute into equation (1):
[tex]\[ 5a\left(\frac{26 - a - ab}{a}\right) = 8(a + \frac{26 - a - ab}{a}) \][/tex]
Simplify and solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- After solving these, the solutions are [tex]\( a = 2 \)[/tex] and [tex]\( b = 8 \)[/tex].
Final Solution: The numbers are 2 and 8.
Thus, given the conditions, the numbers that satisfy the given harmonic mean, arithmetic mean, geometric mean, and the equation [tex]\( 2A + G^2 = 26 \)[/tex] are:
[tex]\[ \boxed{2 \text{ and } 8} \][/tex]