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Sum of n Terms

If a , b , c ...

Question

If $a, b, c$ and $d$ are in G.P., then the value of $(a - c)^{2} + (b - c)^{2} + (b - d)^{2} - (a - d)^{2}$ is

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Solution

Given : $a, b, c$ and $d$ are in G.P.
So,
$\frac{a}{b} = \frac{b}{c} = \frac{c}{d} \dots (1)$
Now,
$(a - c)^{2} + (b - c)^{2} + (b - d)^{2} - (a - d)^{2} = - 2 a c - 2 b c - 2 b d + 2 a d + 2 b^{2} + 2 c^{2}$
Using equation $(1)$ ,
$= - 2 b^{2} - 2 b c - 2 c^{2} + 2 b c + 2 b^{2} + 2 c^{2} = 0$

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