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]
