1 .
的反函数是
,
,设
,且对于
,有
.求
的解析表达式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec11283c9e3c6ea5379370135fcd6818.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bc6c4e5bb6e207512630b32f27db1c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f66a88cbaed58dcf9671ba9240359b9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0316344ece953782b3e3ed27185ba53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7afeb1c6e47de3f6eb82fd4a0234b557.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e338b64edbf3cab8609f51af26c1dd9.png)
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2 . 已知数列
满足
.
(1)若
,证明:
(i)当
时,有
;
(ii)当
时,有
.
(2)若
,证明:当
时,有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a229ee29685c7ec745dcb07717478885.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86fc336b4a83bf6d66c4afcc431597f8.png)
(i)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/030f5b94af92e8995f3ba989f1a03d03.png)
(ii)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e167b43045b3297248e334c41c621b8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5930c0f0a7a529f37be1952d976401bd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5d0a73f50b3e4583f1c1b6d6bf0d18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e78a219a67a184a4307954aeeb33b4b.png)
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3 . 数列
满足:
,
,(
表示不大于x的最大整数,
).试求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/489340c9a2d70c00bae13b7018cad448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b62f3fd059c5c33802cb98123374f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f73940d577a0bf4cfd373f0312b75c54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f8ced5ff7dfbe12a7fc3378a5e3c3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17f78969d9ac865772e51b9b5514868e.png)
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4 . 将长为
、宽为
的矩形划分为
个小正方形.一粒子不重复不遗漏连续地通过每个小正方形的一条对角线.这件事能否办到?若办不到,请说明理由;若能办到,请给出一种行走路线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e70354d6ca5ad9f6b4592fac0b5e559.png)
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5 . 一个人上台阶可以一次上1级台阶,也可以一次上3级台阶,或者一次上4级台阶.若这个人上
级台阶总共有
种走法,证明
为平方数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2bfe05ef6780e5f032ca3a7c96811a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b07f7a46323e7630dd8cd5cffcb11a5d.png)
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6 . 求集合
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b96ab3cc4f183937d0ecf24feeff9508.png)
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7 . 设
,其中,b为正奇数.定义数列
满足
,
.若正整数
,使得
为素数.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a29fca134ddbac482c9821e78e6ab842.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1f83d1a6f169d2e621b366431729888.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91fc7b499561a3427ed1ef0354689f7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/772e2d1d7398351c44c55366756f16ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53500b98cf6e4620e77a8271c73549d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76a3ac74cdb64183be40d576145a9cd5.png)
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8 . 已知数列
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b81341db9a71901aadde68f5dfbf2024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1254070260067f8bf2fec39a7d0c8f1.png)
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9 . 已知
,
.求证:对一切
,
,…,
,均有
,等号当且仅当
时成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be466586da8810ccfd811c59a747adb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf10e7b0809be29347785970158557fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a774356dc406f4f2c2b5295a65bee96f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e60c52127beaf19b211d27e3373192c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2697ef67790838c84cc238a0334c5d47.png)
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