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Sum of Infinite Terms of a GP

If r>1 and ...

Question

If $r > 1$ and $x = a + a / r + a / r^{2} + . . ., y = b + b / r + b / r^{2} + . . .$ and $z = c + c / r + c / r^{2} + . . .,$ then value of $x y / z^{2}$ is

A

a b / c^{2}

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B

a b r / c

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C

a b / c^{2} r

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D

none of these

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Solution

The correct option is A $a b / c^{2}$
x,y,z are the sums of three different infinite G.P.'s

$x = a r / (1 - r)$
$y = b r / (1 - r)$
$z = c r / (1 - r)$
$∴ x y / z^{2} = a b / c^{2}$

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z = c + \frac{c}{r^{2}} + \frac{c}{r^{4}} + . . . .

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| r | > 1,

then the value of

\frac{x y}{z}

\frac{p}{a} + \frac{q}{b} + \frac{r}{c} = 1

\frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 0

\frac{p^{2}}{a^{2}} + \frac{q^{2}}{b^{2}} + \frac{r^{2}}{c^{2}}

a, b, c

are all different and the points

(\frac{r^{3}}{r - 1}, \frac{r^{2} - 3}{r - 1})

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are collinear, then show

3 (a + b + c) = a b + b c + c a - a b c .

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