Summation by using the following common summation formulas is the most basic and important method for the summation of series.

1, sum formula of arithmetic sequence:

2. Sum formula of equal ratio series:

Power sum formula of natural numbers:

3、 4、

5、

[example] sum1+x2+x4+X6+… x2n+4 (x ≠ 0)

∴ This series is a geometric series with the first term 1 and the common ratio of x2, and it has n+3 terms.

When x2 = 1, that is, X = 1, the sum is n+3.

Comments:

(1) Use the summation formula of proportional sequence. When the common ratio is expressed in letters, whether it is 1 is discussed. If this problem is in the form of "equal ratio" and is not specified as a geometric series, then whether x is 0 should also be discussed.

(2) Find the number of items in the series * * *, and the last item is not necessarily the nth item.

Corresponding to the college entrance examination questions: let the top sum of the series 1, (1+2), …, (1+2+), … be the value.

Second, the dislocation subtraction summation

The sum of dislocation subtraction plays a very important role in the college entrance examination. In recent years, the series of college entrance examination questions have covered this aspect. It is necessary for our students to master this method seriously. This method is used to derive the first n terms and formulas of geometric series. This method is mainly used to find the sum of the first n terms of the sequence {an bn}, where {an} and {bn} are arithmetic progression and geometric progression respectively. Summation is generally on both sides of the known summation formula. Then the new sum is subtracted from the original sum and converted into the sum of geometric series with the same multiple, which is called dislocation subtraction.

[Example] Sum:()...................①

It can be seen from the title that the general term of {} is the product of the general term of arithmetic progression {2n- 1} and the general term of geometric series {}.

A set of ................... (2) (setting system dislocation)

①-② (Dislocation Subtraction)

Using the summation formula of equal proportion series, we can get:

∴

Note that 1 is a special case when the common ratio x is 1.

2 Dislocation subtraction should pay attention to the last item

The characteristic of this kind of problem is that the sequence is multiplied by a counterpart of arithmetic progression and geometric progression.

Corresponding to the college entrance examination questions: let the first term of positive geometric series be the sum of the first n terms, (i) find the general term; (ii) the sum of the first n items.

Third, reverse addition.

This is a method to derive the sum formula of the first n terms of arithmetic progression, that is, to arrange a series in reverse order (reverse order) and then add it to the original series to get n. 。

[Example] Verification:

Proof: Hypothesis

Turn the right side of formula ① upside down.

(In reverse order)

There are also

…………..……..②

①+② (sum in reverse order)

∴

Fourth, grouping summation.

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.

If the general formula of a series is, one of them is arithmetic progression and the other is geometric progression, it is generally summed by grouping combination method.

[Example]: Find the sum of the first n items of a series;

Analysis: the general formula of the series is, while the series is arithmetic progression and geometric progression, which are generally summed by grouping and combination method;

[Solution]: Because, so.

(grouping)

The first bracket is the sum of geometric progression, and the last bracket is the sum of arithmetic progression. Therefore,

Verb (verb's abbreviation) Split Term Sum Method

This is the concrete application of the idea of decomposition and combination in the summation of series. The essence of the split term method is to decompose each term (general term) in the series, and then recombine it, so that it can eliminate some terms and finally achieve the purpose of summation. The general term decomposition (split term) is as follows:

( 1) (2)

(3) (4)

(5)

Find the sum of the first n terms of a sequence.

Hypothesis (split term)

Then (sum of split items)

=

=

Summary: This deformation is characterized by splitting each item in the original series into two items, and most of the items in the middle cancel each other out, leaving only a few items.

Note: The remaining projects have the following characteristics.

1 The position of the remaining items is symmetrical before and after.

The positive and negative of the other items are opposite.

[Exercise] In the sequence {an}, find the sum of the first n items in the sequence {bn}.