csat-suneung 2006 Q13

csat-suneung · South-Korea · csat__math-science 4 marks Arithmetic Sequences and Series Properties of AP Terms under Transformation
Two sequences $\left\{ a _ { n } \right\} , \left\{ b _ { n } \right\}$ are given by
$$\begin{aligned} & a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } \cos \frac { ( n - 1 ) \pi } { 2 } \\ & b _ { n } = \frac { 1 + ( - 1 ) ^ { n - 1 } } { 2 ^ { n } } \end{aligned}$$
Which of the following statements in are true? [4 points]

ㄱ. For all natural numbers $k$, $a _ { 3 k } < 0$. ㄴ. For all natural numbers $k$, $a _ { 4 k - 1 } + b _ { 4 k - 1 } = 0$. ㄷ. $\sum _ { n = 1 } ^ { \infty } a _ { n } = \frac { 3 } { 5 } \sum _ { n = 1 } ^ { \infty } b _ { n }$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ
Two sequences $\left\{ a _ { n } \right\} , \left\{ b _ { n } \right\}$ are given by

$$\begin{aligned}
& a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } \cos \frac { ( n - 1 ) \pi } { 2 } \\
& b _ { n } = \frac { 1 + ( - 1 ) ^ { n - 1 } } { 2 ^ { n } }
\end{aligned}$$

Which of the following statements in <Remarks> are true? [4 points]

<Remarks>

\noindent ㄱ. For all natural numbers $k$, $a _ { 3 k } < 0$.\\
ㄴ. For all natural numbers $k$, $a _ { 4 k - 1 } + b _ { 4 k - 1 } = 0$.\\
ㄷ. $\sum _ { n = 1 } ^ { \infty } a _ { n } = \frac { 3 } { 5 } \sum _ { n = 1 } ^ { \infty } b _ { n }$\\
(1) ㄱ\\
(2) ㄴ\\
(3) ㄷ\\
(4) ㄱ, ㄴ\\
(5) ㄴ, ㄷ