발산 수열 비의 극한

$c _{\mathit{n}} = 2 a _{n} - 5 b _{n}$이라 하면 $\displaystyle b _{\mathit{n}} = \frac{1}{5} \left( 2 a _{n} - c _{n} \right)$ $\displaystyle \lim\limits _{n \rightarrow \infty} {} {c _{n}} = 3$이므로 $\displaystyle \lim\limits _{n \rightarrow \infty} {} \frac{c _{n}}{a _{n}} = 0$ $\displaystyle \lim\limits _{n \rightarrow \infty} {} \frac{2 a _{n} + 3 b _{n}}{a _{n} + b _{n}}$$\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{2 a _{n} + 3 \cdot \frac{1}{5} \left( 2 a _{n} - c _{n} \right)}{a _{n} + \frac{1}{5} \left( 2 a _{n} - c _{n} \right)}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{16 a _{n} - 3 c _{n}}{7 a _{n} - c _{n}}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{16 - 3 \cdot \frac{c _{n}}{a _{n}}}{7 - \frac{c _{n}}{a _{n}}}$ $\displaystyle = \frac{16}{7}$ $\therefore$ $p + q = 7 + 16 = 23$ [다른풀이] $\displaystyle \lim\limits _{n\rightarrow \infty} ( 2 a _{n} - 5 b _{n} ) = 3$이므로 $\displaystyle \lim\limits _{n \rightarrow \infty} {} a _{n} \left( 2 - 5 \cdot \frac{b _{n}}{a _{n}} \right) = 3$에서 $\displaystyle \lim\limits _{n\rightarrow \infty} a _{n}$$= \infty$ 이므로 $\displaystyle \lim\limits _{n \rightarrow \infty} {} \left( 2 - 5 \cdot \frac{b _{n}}{a _{n}} \right) = 0$ $\therefore$ $\displaystyle \lim\limits _{n\rightarrow \infty} \frac{b _{n}}{a _{n}} = \frac{2}{5}$ $\displaystyle \lim\limits _{n\rightarrow \infty} \frac{2 a _{n} + 3 b _{n}}{a _{n} + b _{n}} = \lim\limits _{n\rightarrow \infty} \frac{2 + 3 \cdot \frac{b _{n}}{a _{n}}}{1 + \frac{b _{n}}{a _{n}}}$$\displaystyle = \frac{2 + 3 \cdot \frac{2}{5}}{1 + \frac{2}{5}} = \frac{16}{7}$ $\therefore$ $p + q = 7 + 16 = 23$