Solution: Let the sequence 1, 5,3,10,5,15,7, (), (), 25, 1 1 be a 1, b/kloc.

Then we can get two subsequences a 1, a2, a3 and b 1, b2, b3.

Then a 1+2=a2, a2+2=a3, a3+2=a4, that is, a(n- 1)+2=an,

Then a5=a4+2=7+2=9,

Similarly, b 1= 1*5, b2=2*5, b3=3*5, that is, bn=n*5,

You can get b4=4x5=20.

Then the order is 1, 5,3,10,5,15,7, (20), (9), 25, 1 1.

Sequence classification

Sequence can be divided into finite sequence and infinite sequence, periodic sequence and constant sequence.

series formula

(1) general term formula

The relationship between the nth an of a series and the ordinal n of this series can be expressed by a formula an=f(n), which is called the general term formula of this series.

For example: an=3n+2

(2) Recursive formula

If the relationship between the nth term of series an and its previous term or terms can be expressed by a formula, then this formula is called the recursive formula of this series.

For example: an=a(n- 1)+a(n-2)

Properties of arithmetic series

An important condition for (1) series to be arithmetic progression is that the sum of the first n terms of the series can be written as s = an 2+bn.

(2) The sequence a(n+ 1)-an=d(d is a constant) is equivalent to the sequence An being arithmetic progression.

(3) The formula of the sum of the top n items in arithmetic progression is Sn = n * a1+n * (n-1) * d/2.