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The sum of the intercepts cut off by the axes on lines $x + y = a, x + y = ar, x + y = {ar}^{2} \dots \dots$
where $a \neq 0$ and $r = \frac{1}{2}$

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$a \sqrt{2}$

$2 \sqrt{2 a}$

answer is C.

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Detailed Solution

$\begin{array}{rcl} A (a, 0), B (0, a) \\ A_{1} B_{1} & = & = \sqrt{2} a, A_{2} B_{2} = \sqrt{2} ar \\ sum & = & \sqrt{2} a (1 + r + r^{2} + \dots) \\ = & \sqrt{2} a (\frac{1}{1 - r}) \\ = & \frac{\sqrt{2} a}{1 / 2} = 2 \sqrt{2} a \end{array}$

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