Formula method of summation formula of arithmetic sequence
an=a 1+(n- 1)d .
The first n terms and formulas are: Sn=na 1+n(n- 1)d/2.
If the tolerance d =1:sn = (a1+an) n/2;
If m+n=p+q, then: AM+An = AP+AQ exists;
If m+n=2p, then: am+an=2ap.
All the above n are positive integers.
Reverse addition
This is a method to derive the sum formula of the first n terms of arithmetic progression, that is, arrange a series in reverse order (reverse order) and then add it to the original series to get n (a 1+an).
Sn = a1+A2+A3+...+An.
Sn =an+ an- 1+an-2......+a 1 .
Add up and down to get Sn=(a 1+an)n/2.
Grouping method
There is a series, which is neither arithmetic progression nor geometric progression. If this kind of series is properly decomposed, it can be divided into several arithmetic progression, proportional series or ordinary series, and then summed and merged separately.
For example: an=2n+n- 1, which can be regarded as the sum of 2n and n- 1;
Sn=a 1+a2+...+ An
=2+0+22+ 1+23+2+...+2n+n- 1
=(2+22+...+2n)+(0+ 1+...+n- 1)
= 2(2n- 1)/(2- 1)+(0+n- 1)n/2
=2n+ 1+n(n- 1)/2-2
Extended reading: the observation method of mathematics learning review method
Observation method is a problem-solving method by observing the changing law and position characteristics of numbers in the topic, the relationship between conditions and conclusions, the structural characteristics and graphic characteristics of the topic, so as to find out the quantitative relationship in the topic and solve the problem stage. Observe in order, seriously and truly, use your head in observation, think out the truth and find out the law.
Hypothetical method
When we encounter some problems with few conditions and no way to start, we can assume some simple and easy-to-calculate quantities, or assume the problem of motion change, or static special problems; For a topic with many conditions that cannot be sorted out, assume that several different conditions are the same, and so on. This will break through the imprisonment of conventional thinking and get a clever solution, which is also a flexible and extreme strategy.
algebraic method
When solving mathematical problems, letters are used instead of unknowns, and equations are listed according to the equivalence relation, thus the results are obtained. This method is called algebraic method. Learning to solve problems with algebra is like mastering the golden key to solving problems.
Plastic combination
In a very interesting mathematics subject, "number" and "shape" are like inseparable brothers, and almost all quantitative relations or mathematical laws can be reflected by intuitive diagrams. As Hua, a famous mathematician, said, "It is less intuitive to count missing shapes, and it is difficult to count missing shapes." If we can use the strategic analysis of the combination of number and shape to solve the problem, we will give full play to the mutual assistance of number and shape, and make the problem very intuitive, easy to understand and self-evident.
Reverse reasoning method
Everyone knows the story of Sima Guang smashing a jar. Generally speaking, people are fished out of the tank, that is, people leave the water, but it is time-consuming and laborious to fish people, and dare not delay time. Smart Sima Guang thinks from the opposite side that it is too simple to let water leave people-smashing cans. Mathematically, this method is called inverse deduction, also called reduction, that is, inverse deduction from the final result, which is a method to solve mathematical problems.