If between any two quantities there be inserted two arithmetic means A1 and A2 ; two geometric means G1 and G2 and two harmonic means H1 and H2 then prove that G1G2:H1H2=A1+A2:H1+H2
quantities meaning like a set of numbbers?? y= {a1,a2,a3,a3....an}
two of them
ohh ok
AP : \[x, A1, A2, y\] \(\large \implies A1 - x = y - A2~~~~~~~~\color{Red}{(1)} \) GP : \[x, G1, G2, y\] \(\large \implies G1G2 = xy~~~~~~~~\color{Red}{(2)} \) HP : \[x, H1, H2, y\] \(\large \implies \dfrac{1}{H1} - \dfrac{1}{x} = \dfrac{1}{y} - \dfrac{1}{H2}~~~~~~~~\color{Red}{(3)} \)
eliminate x, y from above 3 equations ^^
eliminate meaning
Notice that we have invented x, y variables they don't exist anywhere in the given problem
lets try this : from \(\large \color{Red}{(3)}\) : \[\large \dfrac{1}{H1} - \dfrac{1}{x} = \dfrac{1}{y} - \dfrac{1}{H2}\] rearragingin/simplifying gives you : \[\large \dfrac{H1+H2}{H1H2} = \dfrac{x+y}{xy}\]
use first two equations to substitute x+y and xy values
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