수열 극한 성질

[출제의도] 수열의 극한의 성질을 이해한다. $a _{n} + 2 b _{n} = c _{n}$ …… ㉠, $2 a _{n} + b _{n} = d _{n}$ …… ㉡이라 하면 $\displaystyle \lim\limits _{n \rightarrow \infty} {c _{n}}$$\displaystyle = \lim\limits _{n \rightarrow \infty} {( a _{n} + 2 b _{n} )}$$= 9$, $\displaystyle \lim\limits _{n \rightarrow \infty} {d _{n}}$$\displaystyle = \lim\limits _{n \rightarrow \infty} {( 2 a _{n} + b _{n} )}$$= 90$ $2 \times \text{㉡} - \text{㉠}$에서 $3 a _{n} = 2 d _{n} - c _{n}$이므로 $\displaystyle a _{n} = \frac{1}{3} ( 2 d _{n} - c _{n} )$ $2 \times \text{㉠} - \text{㉡}$에서 $3 b _{n} = 2 c _{n} - d _{n}$이므로 $\displaystyle b _{n} = \frac{1}{3} ( 2 c _{n} - d _{n} )$ 따라서 $a _{n} + b _{n}$$\displaystyle = \frac{1}{3} ( c _{n} + d _{n} )$이므로 $\displaystyle \lim\limits _{x \rightarrow \infty} {( a _{n} + b _{n} )}$$\displaystyle = \lim\limits _{x \rightarrow \infty} {\frac{1}{3} ( c _{n} + d _{n} )}$$\displaystyle = \frac{1}{3} \lim\limits _{x \rightarrow \infty} {( c _{n} + d _{n} )}$ $\displaystyle = \frac{1}{3} \left( \lim\limits _{x \rightarrow \infty} {c _{n} + \lim\limits _{x \rightarrow \infty} {d _{n}} )} \right.$$\displaystyle = \frac{1}{3} \times ( 9 + 90 )$$= 33$ [다른 풀이] $\displaystyle \lim\limits _{n \rightarrow \infty} ( a _{n} + 2 b _{n} ) = 9$, $\displaystyle \lim\limits _{n \rightarrow \infty} {( 2 a _{n} + b _{n} )}$$= 90$이므로 $\displaystyle \lim\limits _{n \rightarrow \infty} ( a _{n} + 2 b _{n} ) + \lim\limits _{n \rightarrow \infty} {( 2 a _{n} + b _{n} )}$$\displaystyle = \lim\limits _{n \rightarrow \infty} \left\{ ( a _{n} + 2 b _{n} ) + ( 2 a _{n} + b _{n} ) \right.$ $\displaystyle = \lim\limits _{n \rightarrow \infty} 3 ( a _{n} + b _{n} )$ 그러므로 $\displaystyle \lim\limits _{n \rightarrow \infty} 3 ( a _{n} + b _{n} )$$= 9 + 90$$= 99$ 따라서 $\displaystyle \lim\limits _{n \rightarrow \infty} ( a _{n} + b _{n} )$$\displaystyle = \frac{1}{3} \lim\limits _{n \rightarrow \infty} 3 ( a _{n} + b _{n} )$$\displaystyle = \frac{1}{3} \times 99$$= 33$