上海市浦东新区2010届高三上学期期末质量抽测数学(理科)试卷2010(3)

①当n为偶数时：cn?cn?21?1??3?(?)n?1??3???2?2?3n?1，cn?2?cn?41?1??3?(?)n?3??3???2?2?n?3，

1?1???c4?c2?3?(?)3??3???，

2?2?所以 cn?c2??c4?c2?????cn?2?cn?4???cn?cn?2?

?1?1??1???3???????????2???2?2?3n?3?1?????2?n?1????11?1????22?4? ??3?11?4n2?1?即cn??2????2?n?1(n?N?)；??????????????????????12分

1?1??3?(?)n?1?3???2?2?n?1②当n为奇数时：cn?cn?2，cn?2?cn?421?1??3?(?)n?3?3???2?2?n?3，

1?1???c3?c1?3?(?)2?3???，

2?2?所以 cn?c1??c3?c1?????cn?2?cn?4???cn?cn?2?

n?1??3??1?2?1???n?3n?1n?14??4????1?3?1??1???1????1?3?????????????1??2???

1?2??2???2???2???1?4?1?即cn?2????2?n?1(n?N?)；???????????????????????14分

n为奇数时

n?1?1???2?????2?综合①②得 cn??n?1?1?????2??2???n为偶数时aa?2n? 22n?1?1?所以 cn?2????2?n?1， bn?n?22aa?2?1?所以f(n)?bn?cn?n?n????22?2?n2?2，????????15分

aa?2?1?则f(n?1)?(n?1)?(n?1)?????2，

22?2?nn?1????aa?21aa?21????22f(n?1)?f(n)??(n?1)?(n?1)?????2???n?n?????2?

22222???2??????????1?a?2n?1???? ????????16分

?2?2n

n????1?a??因为数列?2n?1?????对任意n?N是单调递增数列，且?16?a??14

?2?2????1a15?1?a所以当n?4时，f(n?1)?f(n)?2n?1?????9????0

16216?2?2即f(4)?f(5)?f(6)???f(n)??

?1?a所以当1?n?3时f(n?1)?f(n)?2n?1?????0，

?2?2即f(1)?f(2)?f(3)?f(4)

a?21??2 当n?4时， f(4)?16?2a?28a?215105??2?a?所以f(n)min?f(4)?16?2a? ?????????18分 2828nn

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