gaokao 2015 Q19

gaokao · China · hunan-arts Sequences and series, recurrence and convergence Closed-form expression derivation

19. (This question is worth 13 points) Let $S _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given $a _ { 1 } = 1 , a _ { 2 } = 2$, and $a _ { n + 2 } = 3 S _ { n } - S _ { n + 1 } , n \in \mathbb { N } ^ { * }$. (I) Prove that: $a _ { n + 2 } = 3 a _ { n }$ (II) Find $\mathrm { S } _ { \mathrm { n } }$

19. (This question is worth 13 points)\\
Let $S _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ a _ { n } \right\}$. Given $a _ { 1 } = 1 , a _ { 2 } = 2$, and $a _ { n + 2 } = 3 S _ { n } - S _ { n + 1 } , n \in \mathbb { N } ^ { * }$.\\
(I) Prove that: $a _ { n + 2 } = 3 a _ { n }$\\
(II) Find $\mathrm { S } _ { \mathrm { n } }$

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