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[Solved] Geometric sequence arithmetic sequence fractions

 
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June 15, 2021 11:20 am
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x, y, z  are real numbers, 3x, 4y, 5z  follow a geometric sequence, and  \tfrac{1}{x}, \tfrac{1}{y}, \tfrac{1}{z}   follow an arithmetic sequence, find the value of \tfrac{x}{z}+\tfrac{z}{x} 

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June 15, 2021 11:21 am
Topic starter
Staff 
(@armor)
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  Honorable Member
317 Posts
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\large (4y)^2=15xz   (i)

\large \frac{2}{y}=\frac{1}{x}+\frac{1}{z}    (ii)

(ii)     \large \Rightarrow y= \frac{2xz}{x+z}

substitute it into (i):

\large 16(\tfrac{2xz}{x+z})^2=15xz\Rightarrow \tfrac{(x+z)^2}{xz}=\tfrac{64}{15}

\large \Rightarrow\frac{x}{z}+2+\frac{z}{x}=\frac{64}{15}\Rightarrow \frac{x}{z}+\frac{z}{x}=\frac{34}{15}

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