Q84 / 160 · AP EAMCET 2012 · Mathematics

$\sum_{k=1}^{\infty} \sum_{r=0}^k \frac{1}{3^k}\left({ }^k C_r\right)$ is equal to

  • A
    $\frac{1}{3}$
  • B
    $\frac{2}{3}$
  • C
    $1$
  • Correct answer
    D
    $2$

Solution

$\begin{aligned} & \sum_{k=1}^{\infty} \sum_{r=0}^k \frac{1}{3^k}\left({ }^k C_r\right) \\ = & \sum_{k=1}^{\infty} \frac{1}{3^k}\left({ }^k C_0+{ }^k C_1+{ }^k C_2+\ldots+{ }^k C_k\right) \\ = & \sum_{k=1}^{\infty} \frac{2^k}{3^k}=\sum_{k=1}^{\infty}\left(\frac{2}{3}\right)^k \\ = & \frac{2}{3}+\left(\frac{2}{3}\right)^1+\ldots \infty \\ = & \frac{2 / 3}{1-\frac{2}{3}} \quad\left(\because \text { common ratio }=\frac{2}{3}\right) \\ = & 2\end{aligned}$

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

Relevant Questions