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高中数学学习笔记之数列求和(奇偶项)

  • 2026-03-22 02:55:32
高中数学学习笔记之数列求和(奇偶项)
  高中数学学习中，数列版块的内容和其他版块相比应该是相对简单，在常规考察中：数列的基本量运算、基本性质是高考小题的常客，而大题一般情况下考察数列通项的求解、数列类型的证明以及数列前n项和的求解三类。至于数列与函数、导数或者其他知识点交汇考察，则不在常规考察之类，一旦出现，一般不会出现在前三道大题中，不在此赘述。
  数列求和的常规方法有错位相减法、裂项相消法、分组求和法、并项求和法以及倒序相加法。而奇偶项求和属于子数列问题，其求和的主要方法是分组求和法以及并项求和法，在数列求和问题中属于相对难掌握的问题。
  在奇偶项的实际运算过程中，运算速度慢以及运算准确度较低的因素主要有：
1.分组求和，应用求和公式时，项数未能精确把握，导致运算出错。奇偶项求和过程涉及分类讨论，在总项数n为偶数时，能够比较清晰的认识到奇数项和偶数项各为一半，但到n为奇数时，对于奇数项和偶数项的项数就容易混淆了。对此我们可以通过以下方式进行克服：①利用新元m，对n的奇偶进行刻画，当n为偶数时，设n=2m,那么显然奇数项和偶数项各为m项；当n为奇数时，n=2m+1，此时偶数项为m项，奇数项为m+1项，不过最后需要换元回去②优先讨论n为偶数时情况，得到求和公式，而当n为奇数时，则n+1为偶数，则前n项和等于前n+1项➖️第n+1项，在偶数项通项公式以及求和公式明确的情况下，可总结代入得出答案。
2.在运算过程中，奇数项与偶数项通项公式往往不同，并且可能涉及公差或者公比的中途变化，这使得我们在运算过程中容易出错。
3.有时候，巧用相邻项之和的规律进行并项求和能提高我们的速度；除此之外，奇数项和和偶数项和之间有时也会有规律，此时将他们进行代数转化可以有效节省时间。
解决奇偶项求和问题的准确度以及速度的提升，需要不断的练习，可以进行一题多解探究以及大题思路练习，熟而生巧。
提供一道例题，答题过程仅供个人学习理解，偏繁琐啰嗦，仅供参考。
更多例题，不做过多展示。
本文来自网友投稿或网络内容，如有侵犯您的权益请联系我们删除，联系邮箱：wyl860211@qq.com 。

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  139. /yingpanguazai/ssd/ssd1/www/no.67808.cn/runtime/temp/6df755f970a38e704c5414acbc6e8bcd.php ( 12.06 KB )
  140. /yingpanguazai/ssd/ssd1/www/no.67808.cn/vendor/topthink/think-trace/src/Html.php ( 4.42 KB )
  1. CONNECT:[ UseTime:0.001154s ] mysql:host=127.0.0.1;port=3306;dbname=no_67808;charset=utf8mb4
  2. SHOW FULL COLUMNS FROM `fenlei` [ RunTime:0.001422s ]
  3. SELECT * FROM `fenlei` WHERE `fid` = 0 [ RunTime:0.000712s ]
  4. SELECT * FROM `fenlei` WHERE `fid` = 63 [ RunTime:0.000679s ]
  5. SHOW FULL COLUMNS FROM `set` [ RunTime:0.001396s ]
  6. SELECT * FROM `set` [ RunTime:0.000598s ]
  7. SHOW FULL COLUMNS FROM `article` [ RunTime:0.001722s ]
  8. SELECT * FROM `article` WHERE `id` = 476338 LIMIT 1 [ RunTime:0.000992s ]
  9. UPDATE `article` SET `lasttime` = 1774207599 WHERE `id` = 476338 [ RunTime:0.025050s ]
  10. SELECT * FROM `fenlei` WHERE `id` = 65 LIMIT 1 [ RunTime:0.032800s ]
  11. SELECT * FROM `article` WHERE `id` < 476338 ORDER BY `id` DESC LIMIT 1 [ RunTime:0.011083s ]
  12. SELECT * FROM `article` WHERE `id` > 476338 ORDER BY `id` ASC LIMIT 1 [ RunTime:0.004361s ]
  13. SELECT * FROM `article` WHERE `id` < 476338 ORDER BY `id` DESC LIMIT 10 [ RunTime:0.006483s ]
  14. SELECT * FROM `article` WHERE `id` < 476338 ORDER BY `id` DESC LIMIT 10,10 [ RunTime:0.002291s ]
  15. SELECT * FROM `article` WHERE `id` < 476338 ORDER BY `id` DESC LIMIT 20,10 [ RunTime:0.001648s ]
0.257073s