Progression (Sequence and Series)HardBloom L3
Question
If A, G, H be respectively the A.M., G.M. and H.M. between two positive numbers and if $xA = yG = zH$ where $x,y,z$ are non-zero positive quantities, then $x,y,z$ are in
Options
A.A.P.
B.G.P.
C.H.P.
D.Nothing can be said
Solution
Sol. [Key Concept: Between two positive numbers $A,G,H$ satisfy $G^{2 =}AH$ and $Age\ Gge\ H$ — use to classify the given terms.]
$$\begin{matrix} & & \ \ \ \ \ \ \ \ & xA = yG = zH & \ = \lambda \\ & & \ \ \ \ \ \ \ \ & G^{2} & \ = AH \\ & & \ \ \ \ \ \ \ \ & \left( \frac{\lambda}{y} \right)^{2} & \ = \left( \frac{\lambda}{x} \right)\left( \frac{\lambda}{z} \right) \\ \Rightarrow & & \ \ \ \ \ \ \ \ & y^{2} & \ = xz \end{matrix}$$
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