A series of numbers arranged in a certain order is called a series, and the nth item of the series {an} is expressed by a specific formula (including parameter n), which is called the general formula of this series. Just like the analytic expression of a function, the value of the corresponding an item can be obtained by substituting a specific n value. The solution of the general term formula of sequence is usually obtained by transforming its recursive formula many times.
Geometric progression's general term formula If the first term of geometric progression {an} is a 1 and the common ratio is q, the general term formula of series an is an = A 1Q N- 1.
Note: 1) Because an = A 1Q N- 1, it is q >: 0, q≠ 1, so the images of geometric series are some scattered points on the same exponential function whose abscissa is a natural number.
2) The general term formula {an} of geometric series can also be determined by an = amq n-m formula.
Example: It is known that in the geometric series {an}, a 1= 1 and a2=2, so write its general formula.
Solution: Obviously, its general formula is an = 2 n- 1.
General formula of arithmetic progression If the tolerance of arithmetic progression {an} is d, then an=a 1+(n- 1)d, which is the general formula of arithmetic progression {an}.
Note: 1) Because an=nd+(a 1-d), the image of arithmetic progression is some scattered points on the same straight line whose abscissa is natural series, and the geometric meaning of tolerance d is the slope of the straight line.
2) arithmetic progression's general formula {an} can also be determined by the following formula: ①an=am+(n-m)d, ②am+n=(mam-nan)/(m-n).
3) The tolerance d of arithmetic progression {an} can be determined by the formula d=(an-am)/(n-m).