An A.P.,a G.P.,and a H.P. have the same first | vedclass

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(B) Let $a$ and $b$ be the first and last terms of the three progressions,each having $(2n + 1)$ terms.The middle term of the $A.P.$ is $A = \frac{a +

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$G.P.$

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(B) Let $a$ and $b$ be the first and last terms of the three progressions,each having $(2n + 1)$ terms.The middle term of the $A.P.$ is $A = \frac{a + b}{2}$.The middle term of the $G.P.$ is $G = \sqrt{ab}$.The middle term of the $H.P.$ is $H = \frac{2ab}{a + b}$.We observe that $G^2 = (\sqrt{ab})^2 = ab$ and $A 	imes H = \left(\frac{a + b}{2}ight) 	imes \left(\frac{2ab}{a + b}ight) = ab$.Since $G^2 = A 	imes H$,the middle terms $A, G, H$ are in $G.P.$

An $A.P.$,a $G.P.$,and a $H.P.$ have the same first and last terms and the same odd number of terms. The middle terms of the three series are in

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