If a,;b,,c be in G.P. and a + x,;b + x,;c + x | vedclass

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Question

(b) $a + x,\;b + x,\;c + x$ are in $H.P.$

$ \Rightarrow $ $b + x = \frac{{2(a + x)(c + x)}}{{(a + x) + (c + x)}}$

$ \Rightarrow $ $(b + x)(a + c

Vedclass · Accepted Answer

$b$

Vedclass · Answer

(b) $a + x,\;b + x,\;c + x$ are in $H.P.$

$ \Rightarrow $ $b + x = \frac{{2(a + x)(c + x)}}{{(a + x) + (c + x)}}$

$ \Rightarrow $ $(b + x)(a + c + 2x) = 2(a + x)(c + x)$

$ \Rightarrow $ $(a + c + 2b)x + 2{x^2} + ab + bc$

$= 2ac + 2x(a + c) + 2{x^2}$

$ \Rightarrow $ $x(c + a - 2b) = bc + ab - 2ac$

$ \Rightarrow $ $x(c + a - 2b) = bc + ab - 2{b^2}$ ($\because \;a,\;b,\;c$ are in $G.P.$)

==> $x(c + a - 2b) = b(c + a - 2b)$

==> $x = b$, if $c + a - 2b 
e 0$.

If $a,\;b,\,c$ be in $G.P.$ and $a + x,\;b + x,\;c + x$ in $H.P.$, then the value of $x$ is ($a,\;b,\;c$ are distinct numbers)

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