Q63 / 160 · AP EAMCET 2004 · Mathematics

The value of $\sum_{n=0}^{\infty}\left(\frac{2 i}{3}\right)^n$ is

  • Correct answer
    A
    $\frac{9+6 i}{13}$
  • B
    $\frac{9-6 i}{13}$
  • C
    $9+6 i$
  • D
    $9-6 i$

Solution

$\begin{aligned} & \sum_{n=0}^x\left(\frac{2 i}{3}\right)^n \\ &=1+\left(\frac{2 i}{3}\right)+\left(\frac{2 i}{3}\right)^2+\left(\frac{2 i}{3}\right)^3+\ldots . \\ &=\frac{1}{1-\frac{2 i}{3}}=\frac{3}{3-2 i} \times \frac{3+2 i}{3+2 i}\end{aligned}$

Hence, the correct answer is option: 'A'.

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