Finding the law of the number series, the law of the common number series

The law has been clarified and extra points are required.

1 2 3 4 5 Column n (denoted by the letter a).

Line n.

Denoted by the letter B).

Summary of the rules: 1. Extract the series of numbers in the first row:

1 2 5 10 17 ..Sequence 1

1 3 5 7 ..The difference between the two adjacent terms of sequence 1 is formed by the difference of the number series of equals General term: 2n-1

It is easy to conclude that the general term of sequence 1 is (a-1)2+1 ;

2. Extract the first column of the number series (consider the typesetting, put it horizontally.

1 4 9 16 25 ..Sequence 2

This is easy to get out of: b2;

3. Extract the sequence of numbers composed of numbers at the inflection point (also consider the typesetting, and also put it horizontally.)

4. The first n numbers of the nth column can be formed into an equal difference series of 1, the first term of the nth column is solved by the first rule, and the first n numbers of the nth row can be formed into an equal difference series of -1.

With the above relatively simple rules, the answer is simple.

1 Find the number of 13 from the 10th row and 13 from the left: first calculate the number of the 13th from the left of the first row, and substitute 13 into (a-1)2+1 to get 145. The first 13 of column 13

The number can be formed into a series of equal differences of 1, so as to calculate the "10th row, 13 from the left", that is, the 10th number from the top of the 13th column, 154.

2 (Finding the position of the number "127" in this square requires a general mathematical mastery: estimation.) )

Solution 1: (Using the second rule above) estimate the number that is a rational number after the nearest square to 127, 112 = 121. However 127 is greater than 112, apparently 127 jumps to column 12, and again"The first n numbers in column n can form a series of equal differences of 1", you can calculate the longitudinal position of "127" in 12 columns.

is 6, which is "line 6".

Solution 2: Estimate the integer of a(a-1) closest to 127 (using the third rule above), 12*11=132, so a=12, that is, in 12 columns. Again.

It's done with the rule four, and go back and calculate it yourself.

ps:1 In view of the fourth rule, for the problem type "Finding the number of terms at a given position" (only for this square), if the number of rows is greater than the number of columns, the number of rows is added.

hand, estimate the first term of the row series, and then use the fourth rule to solve the column problem (so just count the columns first, and then count the rows).

2. Estimation also belongs to personal ability, and it is generally easy to estimate within 100, and it should be different if it is intentionally difficult.

3. If there are other good methods, please advise.

4. The typesetting was played very hard.

Treat it as a square.

The nth digit of the first row is (n-1) +1

Therefore, the thirteenth and thirteenth rows of the first horizontal row are 145, and the number of the tenth row is 145 + 9 = 154, and the twelfth count of the first row is 122, 123 ,......There are 6 numbers to 127, that is, the sixth horizontal row and the twelfth vertical row.

Equal Difference Series: Generally, if a series starts from the second term, the difference between each term and its previous term is equal to the same constant, this series is called the equal difference series, and this constant is called the tolerance of the equal difference series, the tolerance is usually represented by the letter d, and the first n term is denoted by sn. The series of equal differences can be abbreviated as:

an=a1+(n-1)d

where n=1 a1=s1; n 2 an=sn-sn-1.

an=kn+b(k,b is constant) Derivation: an=dn+a1-d makes d=k, a1-d=b then an=kn+b.

Proportional sequence: Generally, if the ratio of each term to its predecessor from the second term of a series is equal to the same constant, the sequence is called a proportional sequence. This constant is called the common ratio of the proportional series, and the common ratio is usually represented by the letter q. Proportional sequences can be abbreviated as:

a(n)=a1*q n-1 (where the first term is, the common ratio is q); a(n)=s(n)-s(n-1)(n≥2)。

"Equal sum sequence": In a sequence, if the sum of each term and its next term is the same constant, then the sequence is called the sum of the equal sum sequence, and this constant is called the common sum of the sequence.

For a sequence, if the sum of any of its consecutive k(k 2) terms is equal, we call the sequence an equal sum sequence, and its nature is that it must be a cyclic sequence.

The universal formula for finding the law is as follows:

The first one is a series of equal differences, the difference is 4, so f(n)=5+4(n-1)=4n+1.

The second is also a series of equal differences, the difference is -5, so f(n)=2-5(n-1)=7-5n.

The universal formula is unlikely, the easiest way is to draw the corresponding points in the source system, then look at the approximate distribution of the points, then select the corresponding function, and finally find the specific function according to the numerical value; For example, in these two problems, the point distribution is basically a straight line, and the corresponding function is a primary function, that is, a proportional series, which can be solved by y=ax+b.

Find the meaning of filling in the blanks with regularity

In fact, it is to strengthen the familiarity with the general laws of the number series, although it has many solutions, but the main thing is to develop your ability to find the general laws of the number series and guess the general terms of the number series (that is, the ability to use the incomplete induction method of rock states).

In order to encounter some series that are not easy to find the general term through the general method, you can quickly and accurately guess the general term formula of the number series through the first few terms, and then use mathematical induction or counter-proof method or other methods to prove it, bypass the positive mountain, and quickly get its general formula. Therefore, finding the rules and filling in the blanks will still help us to enhance the solution of some difficult and characteristic number series.

The rule is: take any three numbers in a row, and add the first two numbers to equal the third number.

An term is equal to the sum of the first two items, 1+1=2;Before.

Here's how:

Fibonacci sequence, definition: f0=0, f1=1, fn=f(n-1)+f(n-2)(n>=2, n n*).

Here's how to quickly find the number sequence:

1. The direct method is to write directly from the terms of the known sequence, or by performing algebraic operations on the terms of the known sequence.

2. According to the law of the composition of the number series, the observation and analysis method observes the internal relationship between the items of the number series and the number of terms it corresponds, and writes the expression of the nth term an after appropriate deformation, that is, the general term formula.

3. The undetermined coefficient method is the problem of finding the general term formula, that is, when n=1. You can first set the expression of the nth term an about the variable n, then let n 1 respectively, and take the value of the undetermined coefficient by solving the system of equations, so as to obtain the formula of the general term that meets the requirements.

4. The recursive induction method is based on already.

These numbers are 2 2, 2 3, 2 5, 2 9, 2 17 (where 2 3 represents the cube of 2, i.e. 2*2*2, others are similar).

Let's start with the number series 2,3,5, and the imitation model 9,17

It is easy to find that the latter number is 2 times the previous number minus macro filial piety 1, that is, 3 = 2 * 2-1, 5 = 3 * 2-1, 9 = 5 * 2-1, 17 = 9 * 2-1, so the next number should be 17 * 2-1 = 33, so the last digit of the original number series should be 2 33, that is, 85, 8993, 4592

1. Finding the law is the basic skill of primary school mathematics and middle school mathematics teaching, the purpose is to let students discover, experience, and make the simple arrangement of figures and numbers, through comparison, so as to understand and master the method of finding the law, and cultivate students' preliminary observation, operation, and reasoning ability.

2. Common types of finding patterns:

1) Equal difference series type: the difference between the latter term and the previous term is a constant (the general term is an=a1+(n-1)d). For example: 1, 2, 3, 4 ,..

2) Proportional number: the quotient of the latter term and the previous term is constant (the general term is an=a1*q (n-1)). For example: 1, 2, 4, 8 ,..

3) Perfectly squared: The perfect square of certain integers. For example: 1, 4, 9, 16 ,..

4) Second-order equal difference series: the difference is an equal difference series (the general term is a quadratic function about n, an=an + bn). For example: 1, 3, 6, 10 ,..

5) The difference is proportional series: the difference is the proportional series (the general term is the sum of the first n terms of the proportional series): 1, 3, 7, 15 ,..

1. The number series is actually to find the law, to see a number series, first of all, to see the change law of the number series itself, and the complex number series through the decomposition of the individual, or the merger of multiple items, or through other feasible methods, so that the original law is obvious or transformed into a simple law, such as the difference and other laws that have laws to follow, and finally through the knowledge of the solution.

2. For those equal and proportional series, don't consider shortcuts first, the most practical way is to write out the internal relationship of the series through the most basic existing formula, simplify step by step, and substitute the conditions given by the question step by step, and often the answer will come out naturally.

3. As a person who has experienced the college entrance examination, I think that the number series is often a little related to those exponential logarithms, and the questions often have such a tendency, so the memorization of algebraic formulas is still a little helpful for solving the number series problems.

4, that's pretty much it, of course, the most important point, do more questions, college entrance examination and this kind of thing.

The high school number series law mainly learns the equal difference number series and the proportional number series.

The more complex series of numbers can be simplified or decomposed into equal difference series and proportional series to solve.

The ability to simplify or decompose is the key to solving problems.