If a,;b,;c are in A.P., b,;c,;d are in G.P. a | vedclass

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Question

(c) $a,\;b,\;c$ are in $A.P.$ then $2b = a + c$.....$(i)$

$b,\;c,\;d$ are in $G.P. $ then ${c^2} = bd$....$(ii)$

$c,\;d,\;e$ are in $H.P.$ then

Vedclass · Accepted Answer

$G.P.$

Vedclass · Answer

(c) $a,\;b,\;c$ are in $A.P.$ then $2b = a + c$.....$(i)$

$b,\;c,\;d$ are in $G.P. $ then ${c^2} = bd$....$(ii)$

$c,\;d,\;e$ are in $H.P.$ then $d = \frac{{2ce}}{{c + e}}$....$(iii)$

From $(ii)$, $c^2=bd={(a+c)\over 2}{2ce\over(c+e)}$

$ \Rightarrow $ ${c^2} = \frac{{ace + {c^2}e}}{{c + e}}$

$\Rightarrow {c^3} + {c^2}e = ace + {c^2}e$

$ \Rightarrow $ ${c^3} = ace$

$\Rightarrow {c^2} = ae$

Hence $a,\;c,\;e$ will be in $G.P.$

If $a,\;b,\;c$ are in $A.P.$, $b,\;c,\;d$ are in $G.P.$ and $c,\;d,\;e$ are in $H.P.$, then $a,\;c,\;e$ are in

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