First slide

Arithmetico-Geometric Progression

Question

The sum of infinite series $1 - 3 x + 5 x^{2} - 7 x^{3} + ......... \infty,$ $| x | < 1$ is equal to 2, then $x$ is:

Moderate

A

$\frac{1}{4} (- 5 \pm \sqrt{17})$
B

$- \frac{1}{4} (5 + \sqrt{17})$
C

$(\sqrt{17} - 5) / 4$
D

None of these

Solution

$S = 1 - 3 x + 5 x^{2} - 7 x^{3} + .............$
$x S = x - 3 x^{2} + 5 x^{3} - ........$

$\Rightarrow (1 + x) S = 1 - 2 [x - x^{2} + x^{3} - ........]$

$\Rightarrow S = \frac{1 + x - 2 x}{{(1 + x)}^{2}} = \frac{1 - x}{{(1 + x)}^{2}} = 2$ $[∵ | x | < 1]$

$\Rightarrow$ $2 x^{2} + 5 x + 1 = 0$ $\Rightarrow x = (- 5 \pm \sqrt{17}) / 4$

$\Rightarrow$ $x = (\sqrt{17} - 5) / 4$ as $| \frac{- 5 - \sqrt{17}}{4} | > 1$

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