Which term of the series \(\frac{\sqrt[]{5}}{3},\, \frac{\sqrt[]{5}}{4},\frac{1}{\sqrt[]{5}},\, ...\) is \(\frac{\sqrt{5}}{13}\) ?
Step-by-step Solution:
The given series is: \[ \sqrt{5} \left( \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots \right). \] Hence, \[ \frac{1}{\sqrt{5}}(3, 4, 5, \ldots) \] are in arithmetic progression (AP). Suppose \( \frac{13}{\sqrt{5}} \) is the \( n^{\text{th}} \) term. Then, \[ \frac{13}{\sqrt{5}} = \frac{1}{\sqrt{5}} \left[ 3 + (n-1) \times 1 \right] \Rightarrow n = 11. \]