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高中数学选择性必修 第二册(381题)



若数列 \(\left\{ {{a_n}} \right\}\) 满足 \({a_n} + {a_{n + 1}} + {a_{n + 2}} = 2020\)(\(n \in {{\mathbf{N}}^*}\)) , \({a_{2022}} = 1\),\({a_{2021}} = 2\) ,则 \({a_1} = \) (       )


A.1

B.2

C.3

D.2017


知识点:第四章 数列


参考答案:D


解析:


因为 \({a_n} + {a_{n + 1}} + {a_{n + 2}} = 2020\) ,


所以 \({a_{n + 1}} + {a_{n + 2}} + {a_{n + 3}} = 2020\) ,


两式相减可得, \({a_n} = {a_{n + 3}}\) ,


所以数列\(\left\{ {{a_n}} \right\}\)为以\(3\)为周期的周期数列,


又因为 \({a_{2020}} + {a_{2021}} + {a_{2022}} = 2020\) ,


即 \({a_{2020}} + 2 + 1 = 2020\) ,所以 \({a_{2020}} = 2017\) ,


因为 \(2020 = 673 \times 3 + 1\) ,所以 \({a_{2020}} = {a_1} = 2017\) .


故选:D


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