If a, b, c are in A.P. and a, b, d are in G.P.,


If abc are in A.P. and abd are in G.P., show that a, (a − b), (d − c) are in G.P.


$a, b$ and $c$ are in A.P.

$\therefore 2 b=a+c \quad \ldots \ldots(\mathrm{i})$

Also, $a, b$ and $d$ are in G.P.

$\therefore b^{2}=a d \quad \ldots \ldots$ (ii)

Now, $(a-b)^{2}=a^{2}-2 a b+b^{2}$

$\Rightarrow(a-b)^{2}=a^{2}-a(a+c)+a d \quad$ [Using (i) and (ii)]

$\Rightarrow(a-b)^{2}=a^{2}-a^{2}-a c+a d$

$\Rightarrow(a-b)^{2}=a d-a c$


Therefore, $a,(a-b)$ and $(d-c)$ are in G.P.

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