I. Definition of sequence
A sequence refers to a group of numbers arranged according to a certain law. Generally speaking, each number in a series has a position, which is called the number of items in the series. The first item in the series is called the first item, the second item is called the second item, and so on. Laws in a sequence can be expressed by formulas or recursion.
Second, the nature of the sequence
1. finite sequence and infinite sequence
A finite series refers to a series with limited terms, and an infinite series refers to a series with unlimited terms. Infinite series can be divided into monotonically increasing series, monotonically decreasing series, monotonically decreasing series, monotonically increasing series and rocking series.
2. Arithmetic series and geometric series
Arithmetic progression is a series with equal difference between two adjacent terms in an exponential series, and geometric series is a series with equal ratio between two adjacent terms in an exponential series. Both arithmetic progression and geometric progression have some important properties, such as general term formula and summation formula.
3. Recursive order
Recursive series means that every term in exponential series is a series derived from the previous term. Recursive sequence has some important properties, such as general term formula and summation formula.
Third, the sequence of operation steps
1. Formula for finding the general term
The general term formula is the formula of any term in the exponential column. For arithmetic progression and geometric progression, the general formula can be obtained by calculating the tolerance or common ratio. For recursive sequence, the general term formula can be obtained by finding the general term formula of recursive formula.
2. Sum formula
The summation formula is the summation formula of the first n items in the exponential column. For arithmetic progression and geometric progression, the sum formula can be obtained by finding the general formula of the sum of the first n terms. For recursive series, the summation formula can be obtained by recursive formula and general term formula.
Step 3 find out the number of items
The number of items refers to the number of items in the index column. For arithmetic progression and geometric progression, the number of terms can be obtained by finding the general term formula and the first and last terms. For recursive series, we can use recursive formula and general formula to find the number of terms.
4. Find tolerance and common ratio
Tolerance refers to the difference between two adjacent terms in arithmetic progression, and common ratio refers to the ratio between two adjacent terms in geometric progression. For arithmetic progression and geometric progression, the tolerance and common ratio can be obtained by solving the general formula and the first and second terms.
Fourthly, the application of sequence.
Sequence is widely used in real life. For example, population growth can be described by geometric series; The change of species number can be described by recursive sequence; The fluctuation of stock price can be described by arithmetic progression. The application of sequence also includes finance, physics, engineering and other fields.