General formula:
Sum formula: the number of intermediate terms is a quadratic function without constant terms.
2. Geometric series: starting from the second term, the ratio of each term to its previous term is the same constant, and this series is a geometric series.
General formula:
Sum formula:,,, that is, constant term and term coefficient are opposite.
3. Common general terms and summation methods:
(1) form, easy to sum, method: superposition;
For example:
There are:
(2) When the form is divided by the same, the reciprocal of the structure is arithmetic progression;
For example, arithmetic progression with an error of -2.
(3) Form and method: structure: geometric series;
For example, we can get: using the undetermined coefficient method, that is, the equal ratio and the common ratio are both 2.
(4) Form: structure: geometric series;
(5) the same form of division, into the above situation for construction;
Because, if it is converted to the method of (1), if it is not 1, it is converted to the method of (3).
(6) sum: sum in reverse order, if it has the relevant characteristics of arithmetic progression, the sum after reverse order is a fixed value;
(7) Sum: Dislocation subtraction, which is suitable for the form of a series in which the general term formula is an arithmetic linear function multiplied by an equal proportion, such as:
(8) Sum: the split terms cancel each other, which is suitable for the general term formula in fractional form, and one term is split into two or more differences. Such as:,, etc. ;
(9) Sum: group sum, which is suitable for general items that can be divided into two parts or several parts and are easy to sum, such as.
(10) In addition, the law can be found by finding the first term, but this method is not suitable for solving problems.
4. Relationship with ...:
5. Common properties of arithmetic progression:
(1) If A becomes arithmetic progression, then A is called the arithmetic mean of, and A=
(2) in arithmetic progression, if m+n=p+q, then (m, n, p, q ∈ n);
(3) The item marked arithmetic progression in the lower corner is still arithmetic progression;
(4) The sequence formed by the sum of continuous m terms becomes arithmetic progression.
6. Common properties of geometric series:
(1) If, g is a geometric series, then A is called the mean term of, and G=
(2) In geometric series, if m+n=p+q, then (m, n, p, q ∈N).
(3) The term marked with arithmetic progression in the lower corner is still a geometric series;
(4) The sequence formed by the sum of continuous m terms becomes a geometric series.