1 . 已知数列
中,
,前n项和
满足条件
,计算
,
,
,然后猜想出
的表达式,并用数学归纳法证明你的结论,
某学生的解答如下:当
时,
,即
,
∵
,∴
,
,
.
由此猜想
(
).
①当
时,即
.结论成立.
②假设当
(
)时结论成立,即
成立,则当
时,
∵
,∴
,又
.
∴
是首项为3,公比为
的等比数列.由此得
,这表明,当
时结论也成立.
由①②可知,猜想对任意
都成立.
请判断学生的解答是否正确?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10eb75df1ae9ac69895bb8d23a0acb76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daf464629fa321a6ff7401ab79f07083.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
某学生的解答如下:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/307f7e2a06df238a2e615aaec1700dc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cb084a41cfc6177554b4fd7417e54be.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d176c76ce064f003a854be7d217afc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f9f9cef4d13522b5a44bb5a2b4e6724.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f81b4757e2e42ab974451200b0e15f64.png)
由此猜想
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6a062612f4052dede47b4f6cd00a447.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36b98ef143f8159f3a7dafa1fd2f2370.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c87b351f16728b0023fd63678f8103c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c9a4897630c5cd40ca7bea592fbf4f8.png)
②假设当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ef7ca2b3e8061384501f668e59696a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2397df3279607612ea3cbef101ee0bf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90337a4a4579a2b42c18ed8dbb495604.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce57f0dd77ed43604cb6919b65fa2add.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2712a61ea7f1b0c9e2eda865c7f4202e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d6a6a7d7739cc81edb4b6a6d1d8027c.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e8f3fc66b3fa437e13c708d1fff6de2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba21f3d0cfc86d40e2e06446623ce0.png)
由①②可知,猜想对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36b98ef143f8159f3a7dafa1fd2f2370.png)
请判断学生的解答是否正确?
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2021-09-25更新
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3卷引用:第4章 数列 单元综合检测(重点)(单元培优)-2021-2022学年高二数学课后培优练(苏教版2019选择性必修第一册)
(已下线)第4章 数列 单元综合检测(重点)(单元培优)-2021-2022学年高二数学课后培优练(苏教版2019选择性必修第一册)高中数学解题兵法 第一百十四讲 阅读、迁移沪教版(2020) 选修第一册 精准辅导 第4章 4.4(2)数学归纳法的应用