# Find the limit of the sequence or why it diverges?

## ${a}_{n} = \frac{2 \cdot {3}^{n} + 9 \cdot {5}^{n}}{10 \cdot {3}^{n} + 21 \cdot {5}^{n}}$

Jul 20, 2017

I solved it within minutes of posting the question. The answer is $L = \frac{3}{7}$

#### Explanation:

I initially thought that the 2, 9, 10, and 21 were separate terms so I divided ${5}^{n}$ on all of those. I looked again and realized that was not the case.

$= \frac{\frac{2 \cdot {3}^{n}}{5} ^ n + \frac{9 \cdot {5}^{n}}{5} ^ n}{\frac{10 \cdot {3}^{n}}{5} ^ n + \frac{21 \cdot {5}^{n}}{5} ^ n}$

$= \frac{\left(2 \cdot 0\right) + \left(9 \cdot 1\right)}{\left(10 \cdot 0\right) + \left(21 \cdot 1\right)}$

$= \frac{9}{21} = \frac{3}{7}$