The sum of series sums the numbers arranged according to a certain law. Finding Sn is essentially a general formula for finding Sn, so we should pay attention to understanding its meaning. Common methods include formula method, dislocation subtraction method, anti-addition method, grouping method, split term method, mathematical induction method, general term reduction method and union term summation method. Sequence is an important content of algebra in senior high school, and it is also the basis of learning advanced mathematics. It plays an important role in college entrance examination and various math competitions.

Seven methods of summation of sequence;

1, arithmetic series series:

For arithmetic progression an=a 1+(n- 1)d, where a 1 is the first term, d is the tolerance, and n is the number of terms.

The sum formula is Sn=(n/2)(a 1+an), where Sn represents the sum of series.

2, arithmetic progression summation (difference method):

The difference method can be used to solve the sum of arithmetic progression. The specific steps are as follows: the sequence is inversely subtracted to get a sequence with a tolerance of d, and then multiplied by the number of terms n and divided by 2 to get the sum of the sequence.

3, GeometricSeries (geometric series):

For geometric series BN = a 1 * R (n- 1), where a 1 is the first term, r is the common ratio, and n is the number of terms.

The common ratio r is not equal to 1, and the summation formula is sn = a1* (1-r n)/(1-r).

The common ratio r is equal to 1, and the summation formula is Sn=n*a 1.

4, geometric series summation (multiplication):

Multiplication can be used to solve the sum of geometric series. The specific steps are to multiply the series by the common ratio R, then subtract it from the original series to get a arithmetic progression, and then use the formula of arithmetic progression sum to solve it.

5. Sum of power series:

For power series an = a 1 * r (n- 1), where a 1 is the first term, r is the ratio, and n is the number of terms.

The sum formula is Sn=a 1/( 1-r), where Sn represents the sum of series and the absolute value of r is required to be less than 1.

6, geometric series summation:

Geometric series is the part and sequence of geometric series. For geometric series bn = a 1 * r (n- 1), where a 1 is the first term, r is the common ratio, and n is the number of terms.

The summation formula is sn = a1* (1-r n)/(1-r).

7. Sum of special series:

Some special series have unique summation formulas, such as square series and cubic series. For square series and cubic series, you can sum them directly with a specific formula.

Common application scenarios of sequence summation

1, Mathematical problem solving: Sum of series is the basis for solving many mathematical problems. For example, in probability theory and statistics, the sum of series can be used to calculate probability and expectation; In calculus, the sum of series can be used to approximate the integer value of a function.

2. Finance and investment: Sum of series has important applications in the fields of finance and investment. For example, in the calculation of compound interest, you can use the geometric series summation formula to calculate the future value and compound interest; In annuity calculation, arithmetic progression summation can be used to calculate the final value of time deposit and the present value of annuity.

3. Physics and engineering: Series summation is also widely used in physics and engineering. For example, in dynamics, the sum of series can be used to calculate velocity, acceleration and displacement. In signal processing, the spectrum characteristics of the signal can be analyzed by summation of series.