If the first and (2n - 1)^{th} terms of an A. | vedclass

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(D) Let $\alpha$ and $\beta$ be the first and $(2n - 1)^{th}$ terms of the $A.P.$,$G.P.$,and $H.P.$ respectively.For $A.P.$: The $n^{th}$ term is $a =

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$(a)$ and $(c)$ both

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(D) Let $\alpha$ and $\beta$ be the first and $(2n - 1)^{th}$ terms of the $A.P.$,$G.P.$,and $H.P.$ respectively.For $A.P.$: The $n^{th}$ term is $a = \frac{\alpha + \beta}{2}$ $(i)$For $G.P.$: The $n^{th}$ term is $b = \sqrt{\alpha \beta}$ (ii)For $H.P.$: The $n^{th}$ term is $c = \frac{2\alpha \beta}{\alpha + \beta}$ (iii)From $(i)$,(ii),and (iii),we observe that $a, b, c$ are the Arithmetic Mean,Geometric Mean,and Harmonic Mean of $\alpha$ and $\beta$ respectively.We know that for any two positive numbers $\alpha$ and $\beta$,$A.M. \ge G.M. \ge H.M.$Thus,$a \ge b \ge c$,which matches option $(a)$.Also,$ac = \left(\frac{\alpha + \beta}{2}\right) \left(\frac{2\alpha \beta}{\alpha + \beta}\right) = \alpha \beta = b^2$.Therefore,$ac - b^2 = 0$,which matches option $(c)$.Hence,both $(a)$ and $(c)$ are correct.

If the first and $(2n - 1)^{th}$ terms of an $A.P.$,$G.P.$,and $H.P.$ are equal and their $n^{th}$ terms are respectively $a, b$ and $c$,then:

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