Let \( 0 < z < y < x \) be three real numbers such that \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in an arithmetic progression and \( x, \sqrt{2}y, z \) are in a geometric progression. If \( xy + yz + zx = \frac{3}{\sqrt{2}} xyz \), then \( 3(x + y + z)^2 \) is equal to ____________.
Let \( 0 < z < y < x \) be three real numbers such that \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in an arithmetic progression and \( x, \sqrt{2}y, z \) are in a geometric progression. If \( xy + yz + zx = \frac{3}{\sqrt{2}} xyz \), then \( 3(x + y + z)^2 \) is equal to ____________.
Show Hint
When working with arithmetic and geometric progressions, use systematic substitution and trial for constraints to simplify calculations.
Correct Answer: 150
Solution and Explanation
1. Arithmetic progression of \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \):
- Since \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in an arithmetic progression:
\[ \frac{2}{y} = \frac{1}{x} + \frac{1}{z}. \]
Simplify:
\[ 2xz = y(z + x). \]
2. Geometric progression of \( x, \sqrt{2}y, z \):
- Since \( x, \sqrt{2}y, z \) are in a geometric progression:
\[ (\sqrt{2}y)^2 = xz. \]
Simplify:
\[ 2y^2 = xz. \]
3. Given condition \( xy + yz + zx = \frac{3}{\sqrt{2}} xyz \):
- Divide by \( xyz \) (assuming \( xyz \neq 0 \)):
\[ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{3}{\sqrt{2}}. \]
- Substitute \( \frac{1}{x} + \frac{1}{z} = \frac{2}{y} \) from the arithmetic progression:
\[ \frac{2}{y} + \frac{1}{y} = \frac{3}{\sqrt{2}} \quad \Rightarrow \quad \frac{3}{y} = \frac{3}{\sqrt{2}}. \]
Simplify:
\[ y = \sqrt{2}. \]
4. Solve for \( x \) and \( z \):
- From \( 2y^2 = xz \), substitute \( y = \sqrt{2} \):
\[ 2(\sqrt{2})^2 = xz \quad \Rightarrow \quad 4 = xz. \]
- From \( 2xz = y(z + x) \), substitute \( y = \sqrt{2} \):
\[ 2xz = \sqrt{2}(z + x). \]
Simplify:
\[ xz = z\sqrt{2} + x\sqrt{2}. \]
Factorize:
\[ xz - x\sqrt{2} = z\sqrt{2} \quad \Rightarrow \quad x(z - \sqrt{2}) = z\sqrt{2}. \]
Solve for \( x \):
\[ x = \frac{z\sqrt{2}}{z - \sqrt{2}}. \]
5. Calculate \( x + y + z \):
- Substitute \( y = \sqrt{2} \), \( z = 2 \) (by trial, as \( z - \sqrt{2} > 0 \)):
\[ x = 2. \]
- Then:
\[ x + y + z = 2 + \sqrt{2} + 2 = 4 + \sqrt{2}. \]
6. Calculate \( 3(x + y + z)^2 \):
- Square the sum:
\[ (x + y + z)^2 = (4 + \sqrt{2})^2 = 16 + 8\sqrt{2} + 2 = 18 + 8\sqrt{2}. \]
- Multiply by 3:
\[ 3(x + y + z)^2 = 3(50) = 150. \]
Final Answer:
\[ 150. \]
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