1. observation method: when the first few terms of a series are known, when finding the general term of the series, the given term is generally observed and analyzed to find the law, so as to write the general term of the series according to the law.

Second, the accumulation method:

A recursive sequence of the form an+ 1=an+f(n) (where f(n) is a function of n).

Adding the two sides of the above n- 1 expression respectively, we can get: an = f (n-1)+f (n-2)+…+f (2)+f (1)+a1,(n ≥

① If f(n) is a linear function about n, it can be converted into arithmetic progression summation after accumulation; ?

② If f(n) is an exponential function about n, it can be converted into geometric series summation after accumulation;

③ If f(n) is a quadratic function about n, it can be summed in groups after accumulation;

(4) If f(n) is a fractional function about n, the divisible terms are summed after accumulation.

If the relationship between the nth term an and n of the series {an} can be expressed by a formula, this formula is called the general term formula of the series. The general term of some series can be expressed by two or more formulas. There is also a series with general formula, such as a series composed of all prime numbers.

A series of numbers arranged in a certain order is called a sequence, and each number in the sequence is called an item of this number. Each item is called item 1 (or the first item), item 2, ..., item n, ... A sequence can also be regarded as a function, and its domain is natural number set n (or its finite subset {1, 2, 3, ..., n}), and it corresponds to the independent variable values from small to large.

The general form of the sequence can be written as a 1, a2, ..., a, ..., where an is the nth item of the sequence and can also be abbreviated as {an}.

If the relationship between the nth term an and n of the series {an} can be expressed by a formula, this formula is called the general term formula of the series.