Harmonic mean

If x, y, z form a harmonic progression, then y is the harmonic mean of x and z. Find the harmonic mean of the numbers 6 and 5.

Correct result:

y =  5.4545

Solution:

$$\begin{gathered} x=6 \\ z=5 \\ {x}_1=1\mathrm{/}x=1\mathrm{/}6={\displaystyle \frac{1}{6}}\doteq 0.1667 \\ {x}_3=1\mathrm{/}z=1\mathrm{/}5={\displaystyle \frac{1}{5}}=0.2 \\ d=(x_3-x_1)\mathrm{/}2=(0.2-0.1667)\mathrm{/}2={\displaystyle \frac{1}{60}}\doteq 0.0167 \\ {x}_2=x_1+d=0.1667+0.0167={\displaystyle \frac{11}{60}}\doteq 0.1833 \\ y=1\mathrm{/}{x}_2=1\mathrm{/}0.1833={\displaystyle \frac{60}{11}}=5\frac{5}{11}=5.4545 \end{gathered}$$

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