Reprint: summation formula of proportional series and its derivation process

In this paper, from ： etc. Sum formula of ratio sequence , And its derivation process

One 、 The definition of an equal ratio sequence

If a sequence starts from the second term , The ratio of each term to its previous term is equal to the same constant , Then this sequence is called Geometric series （geometric progression）. This constant is called the constant of an equal ratio sequence Gongbi (common ratio).

Sum up the sequence of equal numbers , There is a formula ：

Record the sequence {an} It's an equal ratio sequence , The common ratio is q, Before that n Items and are Sn, Then there are ：

1. Gongbi q=1 when ,S_n=na₁
2. Gongbi q≠1 when ,S_n=a₁(1-qⁿ)/(1-q)=a₁-a_nq.

Two 、 Derivation of the summation formula of the equal ratio sequence

When the common ratio of the sequence of equal numbers equals 1 And the common ratio is not equal to 1 Before n Terms are different from formulas , therefore , Find the front of an equal ratio sequence n Items often need to be divided “ The common ratio is 1” and “ Common ratio is not 1” The two cases are classified and discussed .

1. When “ The common ratio is 1” when , front n The derivation process of terms and formulas is as follows ：
   

2. When “ Common ratio is not 1” when , front n The derivation process of terms and formulas is as follows ：
   

3、 ... and 、 matters needing attention

Because in the sum formula of the proportional series , The common ratio is equal to 1 And the common ratio is not equal to 1 Before n The term is different from the applicable summation formula , So find the front of the sequence of equal proportions n Xiang Heshi , It is often necessary to determine whether the common ratio is equal to 1 Have a classified discussion .

Please refer to the column for more mathematical basis of artificial intelligence 《 Mathematical fundamentals of artificial intelligence 》.

It's not easy to write , Please support ：

If you can get something from reading this article , Please give me some compliments 、 Comment on 、 Collection , Thank you for your support ！

Pay column on the old ape

1. Paid column 《https://blog.csdn.net/laoyuanpython/category_9607725.html Use PyQt Developing graphical interfaces Python application 》 Special introduction based on Python Of PyQt Basic course of graphical interface development , The corresponding article directory is 《 https://blog.csdn.net/LaoYuanPython/article/details/107580932 Use PyQt Developing graphical interfaces Python Application column Directory 》;
2. Paid column 《https://blog.csdn.net/laoyuanpython/category_10232926.html moviepy Audio and video development column ) Detailed introduction moviepy The class correlation method of audio and video clip synthesis processing and the use of correlation method to process related clip synthesis scene , The corresponding article directory is 《https://blog.csdn.net/LaoYuanPython/article/details/107574583 moviepy Audio and video development column Directory 》;
3. Paid column 《https://blog.csdn.net/laoyuanpython/category_10581071.html OpenCV-Python Difficult problem set for beginners 》 by 《https://blog.csdn.net/laoyuanpython/category_9979286.html OpenCV-Python Graphics and image processing 》 The accompanying column of , It is the author's opinion of OpenCV-Python Some problems encountered in the study of graphics and image processing , The relevant data are basically the results of repeated studies of old apes , be conducive to OpenCV-Python Beginners have a deeper understanding of OpenCV, The corresponding article directory is 《https://blog.csdn.net/LaoYuanPython/article/details/109713407 OpenCV-Python Catalog of difficult problem set for beginners 》
4. Paid column 《https://blog.csdn.net/laoyuanpython/category_10762553.html Python Introduction to reptiles 》 From the perspective of Xiaobai, an Internet front-end developer, this paper introduces the knowledge and skills of crawler development , Including the basics of getting started with crawlers , And crawling CSDN The article information 、 Blogger information 、 Like the article 、 Comments, etc .

The first two columns are suitable for a certain amount of Python Xiaobai readers who are basic but have no relevant knowledge learn , In the third column, please combine 《https://blog.csdn.net/laoyuanpython/category_9979286.html OpenCV-Python Graphics and image processing 》 Learning to use .

For lack of Python Basic colleagues , Through the free column of the old ape 《https://blog.csdn.net/laoyuanpython/category_9831699.html special column ：Python Basic course catalog ） Learn from scratch Python.

If you are interested and willing to support the old ape readers , Welcome to the paid column .

Old ape Python, Learn from the old ape Python！

* ░ Go to the old ape Python Blog Directory https://blog.csdn.net/LaoYuanPython ░