Infinitely decreasing geometric series
a,aq,aq^2……aq^n
Where n is close to positive infinity, q
note:
(1) We call the infinite geometric progression of | q | < 1 infinite recursive geometric progression, and the limit of the sum of the first n terms only exists, but it does not exist when |q|≥ 1 infinite geometric progression.
(2)S is the sum of all terms representing infinite geometric series, which is different from the sum of finite terms in meaning. When s is n→∞, the first n terms and the limit of Sn, that is, S=
S=a/( 1-q)
Algorithm of summation formula of equal proportion sequence
In order to understand the algorithm of infinite decreasing geometric progression summation, we need to introduce the summation formula of proportional sequence first.
Let the first term of a geometric series be a 1, the common ratio be q, and the sum of the first n terms of the series be Sn. When the common ratio is not 1,
sn=a 1+a 1q+a 1q^2+...+a 1q^(n- 1)
Multiply both sides of this formula by the common ratio q at the same time and you get
qSn=a 1q+a 1q^2+...+a 1q^(n- 1)+a 1q^n
Subtract two expressions to get.
( 1-q)sn=a 1-a 1q^n
Therefore, when the common ratio is not 1, the summation formula of equal ratio series is sn = [a1(1-q n)]/(1-q).
For infinite decreasing sequence, the common ratio of the sequence is less than 1. When the above formula shows that n tends to positive infinity, the value in molecular brackets tends to 1, and the sum formula of infinite decreasing series is obtained by taking the limit.
S=a/( 1-q)
Solving methods of multiple choice questions in senior high school mathematics
I. Direct method
Starting directly from the conditions of the topic, using related concepts, properties, theorems, rules and formulas, after strict reasoning and calculation, the conclusion of the topic is drawn.
Second, the special case method
Including the selection of special values, special positions, special functions, special series, special graphs and so on. Choose the options that match the meaning of the question, and substitute or compare the options to determine the answer.
This method is called special value substitution test, and it is a frequently used method.
Third, the combination of numbers and shapes.
Drawing graphics or images can make the information provided by the problem more intuitive and reduce the difficulty of thinking, which is a powerful strategy to solve mathematical problems.
Fourth, valuation judgment
Some problems, which belong to the problem of comparing size or determining position, can avoid wasting time by estimating the value or position because of accurate calculation and strict derivation.
Verb (abbreviation of verb) exclusion method (substitution test method)
Make full use of the characteristics of single choice in multiple-choice questions, that is, there is only one correct choice of information, and eliminate it one by one through analysis, reasoning, calculation and judgment, and finally achieve the goal.
Six, you can also use the limit method, scale method and inquiry induction method.