题目内容
若数列{bn}:对于n∈N*,都有bn+2-bn=d(常数),则称数列{bn}是公差为d的准等差数列.如数列cn:若cn=
,则数列{cn}是公差为8的准等差数列.设数列{an}满足:a1=a,对于n∈N*,都有an+an+1=2n.
(Ⅰ)求证:{an}为准等差数列;
(Ⅱ)求证:{an}的通项公式及前20项和S20.
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(Ⅰ)求证:{an}为准等差数列;
(Ⅱ)求证:{an}的通项公式及前20项和S20.
(I)∵数列{an}满足:a1=a,对于n∈N*,都有an+an+1=2n,∴an+1+an+2=2(n+1),
∴an+2-an=2.
∴数列{an}是公差为2的准等差数列.
(II)∵an+an+1=2n,
∴S20=(a1+a2)+(a3+a4)+…+(a19+a20)
=2(1+3+…+19)
=2×
=200.
∴an+2-an=2.
∴数列{an}是公差为2的准等差数列.
(II)∵an+an+1=2n,
∴S20=(a1+a2)+(a3+a4)+…+(a19+a20)
=2(1+3+…+19)
=2×
10×(1+19) |
2 |
=200.
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