I. arithmetic progression
Arithmetic progression is a common sequence, which is characterized in that the difference between each term and its previous term is equal to the same constant. This constant is called tolerance, which is usually represented by the letter d, the first term is represented by a 1, and the general term formula is an=a 1+(n- 1)d, where n represents the number of terms.
The formula of the sum of the top n terms in arithmetic progression is Sn = a1n+[n (n-1) d]/2 or Sn=[n(a 1+an)]/2, where sn represents the sum of the top n terms. Arithmetic progression is widely used in mathematics, statistics, economics, physics and other fields.
Second, geometric series
Geometric series is a common series, which is characterized in that the quotient of each term and its previous term is equal to the same constant. The general term formula of geometric series is an = a 1q (n- 1), and the sum formula of the first n terms is sn = (a1-anq)/[1-q], where q is the common ratio.
Third, Fibonacci sequence
Fibonacci sequence is a special kind of sequence, which is characterized in that each term is the sum of the first two terms. The first few terms of Fibonacci sequence are 0, 1, 1, 2, 3, 5, 8, 13 and so on. Fibonacci sequence is widely used in nature and human life, such as plant growth and animal reproduction.
Properties of arithmetic series
I. Average value
In arithmetic progression, the arithmetic mean of any two terms is equal to the arithmetic mean of these two terms.
Second, the number of items is the same.
In arithmetic progression, if the sum of the corresponding items of two series with the same number of items is equal, then these two series are arithmetic progression.
Third, the first and the last item
In arithmetic progression, the sum of the first item and the last item is equal to the sum of the second item and the penultimate item, and so on.
Fourth, tolerance.
In arithmetic progression, the difference between any term and its previous term is equal to the same constant, and this constant is the tolerance.
Verb (abbreviation for verb) common ratio
In arithmetic progression, the quotient of any term and its previous term is equal to a constant, and this constant is the common ratio.
Six, square
In arithmetic progression, the sum of the squares of any term and its previous term is equal to a constant, which is equal to the square of the first term plus the square of the last term.
Seven. constant
In arithmetic progression, the sum of the root numbers of any term and its predecessor is equal to a constant, which is equal to the root number of the first term plus the root number of the last term.