The concept of 1. 1 sequence

Class goal 1. Understand sequence and related concepts; 2. Understand the general formula of the series and write any item of the series with the general formula; 3. For a relatively simple series, its general term formula will be written according to its first n terms.

1. Generally, a series of numbers arranged in a certain _ _ _ _ is called a series, and each number in the series is called an item of this series. The general form of the sequence can be written as a 1, a2, a3, …, an, … abbreviated as the sequence {an}, in which the first 1 item of the sequence. An is the nth term of a series, also called the general term of a series.

2. The series with limited projects is called _ _ _ _ _ series, and the series with unlimited projects is called _ _ _ _ _ series.

3. If the relationship between the nth item of the series {an} and the serial number n can be expressed by a formula, then this formula is called the _ _ _ _ _ _ _ _ formula of this series.

First, multiple choice questions

1. The general formula of sequence 2, 3, 4, 5, ... is ()

an=n B.an=n+ 1

C.an=n+2 D.an=2n

2. The general term formula of a given series {an} is an =, and the first four terms of the series are () in turn.

A. 1，0， 1，0 B.0， 1，0， 1

C.，0，，0 D.2，0，2，0

3. If the first four terms of a series are 1, 0, 1, 0, then the general term formula of this series cannot be ().

a . an =[ 1+(- 1)n- 1]

B.an=[ 1-cos(n 180 )]

C.an=sin2(n 90)

d . an =(n- 1)(n-2)+[ 1+(- 1)n- 1]

4. Given that the general formula of series {an} is an = N2-n-50, then -8 is () of the series.

A. Item 5 B. Item 6

C. Item 7 D. None

5. The general formula of sequence 1, 3,6, 10, ... is ()

A.an=n2-n+ 1 B.an=

C.an= D.an=n2+ 1

6. Let an =++(n ∈ n++), then an+ 1-an is equal to ().

A.B.

C.+d-

Second, fill in the blanks

7. The general term formula of known series {an} is an =. Then its first four items are _ _ _ _.

8. It is known that the general term formula of the series {an} is an = (n ∈ n+), so it is the _ _ _ _ item of this series.

9. Make a triangle with matchsticks according to the picture below:

According to the law shown in the figure, the relationship between the number of matchsticks an and the number of triangles n can be _ _ _ _ _ _ _ _ _.

10. It is said that mathematicians of Pythagoras (about 570 BC-500 BC) school in ancient Greece often study mathematical problems on the beach. They draw dots on the beach or use pebbles to represent numbers. For example, if they put pebbles into a triangle as shown in the figure, then the corresponding number of pebbles is called triangle number, and the 10 th triangle number is _ _.

Third, answer questions.

1 1. According to the first term of the series, write the general term formula of the following series:

( 1)- 1,7,- 13, 19,… (2)0.8,0.88,0.888,…

(3),,-,,-,,…

(4), 1,,,… (5)0, 1,0, 1,…

12. Known series;

(1) Find the 10 item of this series;

(2) Is it a project in this series? Why?

(3) Verification: All items in the sequence are within the interval (0, 1);

(4) Are there countless columns of items in the interval? If so, how many? If not, explain why.

Ability improvement

13. The general formula of the series A, B, A, B, … is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

14. Try to guess how many points there are in the nth graph according to the following five graphs and the changing rules of corresponding points.

1. Compared with the properties of the elements in the collection, the items in the sequence also have three properties:

(1) Certainty: Whether a number is in a series, that is, whether a number is an item in a series is certain.

(2) Repeatability: Numbers in the series can be repeated.

(3) Orderliness: A series is not only related to the numbers that make up the series, but also related to the arrangement order of these numbers.

2. Not all series can write its general formula. For example, different approximations of π can form a sequence of 3, 3. 1, 3. 14, 3. 14 1, ... according to accuracy, which has no general formula.

3. If a series has a general formula, its general formula can have many forms. For example, the general formula of series-1, 1,-1, 1, … can be written as an = (.

An = where k ∈ n+.

Functional characteristics of 1.2 sequence

Class goal 1. Understand the recurrence formula of sequence, and make clear the similarities and differences between recurrence formula and general term formula; 2. Write the first item of the sequence according to the recurrence formula of the sequence; 3. Understand the relationship between sequence and function, and learn sequence from the viewpoint of function.

1. If the term 1 or the first few terms of the sequence {an} are known, and the relationship between any term an of the sequence {an} and its previous term an- 1 (or the first few terms) can be expressed by a formula, this formula is called the recursive formula of the sequence.

2. The sequence can be regarded as a domain of _ _ _ _ _ _ _ _ (or its finite subset {1, 2,3, ..., n}). When the value of the independent variable changes from small to large, the corresponding column _ _ _ _ _ _ _ _.

3. Generally speaking, if every item from _ _ _ _ _ _ _ is greater than the previous item, that is, _ _ _ _ _ _ _ _, a series {an} is called an increasing series. If each item of _ _ _ _ _ is less than the previous item,

First, multiple choice questions

1. Given an+ 1-an-3 = 0, the sequence {an} is ().

A. Increasing sequence B. Decreasing sequence

C. Constant term D. Uncertainty

2. The recurrence formula of the sequence 1, 3,6, 10, 15, ... is ().

A.an+ 1=an+n，n∈N+

B.an=an- 1+n，n∈N+，n≥2

C.an+ 1=an+(n+ 1)，n∈N+，n≥2

D.an=an- 1+(n- 1)，n∈N+，n≥2

3. It is known that the first term of the series {an} is a 1 = 1, and an+ 1 = an+, then the fourth term of this series is ().

A. 1 B。

C.D.

4. In the sequence {an}, A 1 = 1, for all n≥2, A 1 A2 A3...An = N2, then: A3+A5 equals ().

A.B.

C.D.

5. It is known that the sequence {an} satisfies an+ 1 = If A 1 =, the value of a2 0 10 is ().

A.B.

C.D.

6. If an = is known, the largest and smallest items in the top 30 items of this series are () respectively.

A.a 1，a30 B.a 1，a9

C.a 10，a9 D.a 10，a30

Second, fill in the blanks

7. It is known that the sum of the first n items of the sequence {an} is Sn, a 1 = 3, 4Sn = 6an-an-1+4Sn-1,then an = _ _ _ _ _ _

8. If the known sequence {an} satisfies: a 1 = a2 = 1, an+2 = an+ 1+an, (n ∈ n+), then an >；; The minimum value of n in 100 is _ _ _ _ _.

9. If the sequence {an} satisfies: a 1 = 1 and = (n ∈ n+), then when n≥2, an = _ _ _ _ _ _

10. If the known sequence {an} satisfies: an ≤ an+ 1, an = N2+λ n, n ∈ n+, then the minimum value of real number λ is _ _ _ _ _.

Third, answer questions.

1 1. In the sequence {an}, a 1 =, an = 1-(n ≥ 2, n ∈ n+).

(1) Verification: an+3 = an；;

(2) Find a2 0 10.

12. Given an= (n∈N+ ∈ n+), is there the largest term in series {an}? If yes, find the maximum term; If not, explain why.

Ability improvement

13. If the known sequence {an} satisfies a1=-kloc-0/,an+ 1 = an+, n ∈ n+, then the general formula an = _ _ _ _ _ _

14. Let {an} be a positive sequence with the first term 1, and (n+1) a-na+an+1an = 0 (n =1,2,3, ...

The connection and difference between function and sequence

On the one hand, the sequence is a special function, so when solving the sequence problem, we should be good at using the knowledge, viewpoint and thinking method of the function to solve the problem, that is, using * * * to solve the special problem.

On the other hand, we should also pay attention to the particularity of the sequence (discrete type), because its domain is n+ or its subset {1, 2, ..., n}, so its image is a series of isolated points, unlike the elementary functions we learned earlier, which are generally continuous curves. Therefore, we should make full use of this particularity when solving problems, for example, when studying monotonicity with series. An- 1), the image shows an upward trend, that is, the series increases, that is, {an} increases? an+ 1 & gt； It holds for any n (n∈N+ ∈ n+). Similarly, there is {an} decline? an+ 1 & lt； An holds for any n (n ∈ n+).

1 numeric column

The concept of 1. 1 sequence

answer

Knowledge carding

1. Order 2. Infinite 3. sum up

work design

1.B2

3.d[ Let n = 1, 2,3,4 be substituted for verification. ]

4.C [N2-N-50 =-8, and get n = 7 or n =-6 (not counting). ]

5.C[ Let n = 1, 2, 3, 4, substitute the test of A, B, C, D, exclude A, B, D and choose C]

6.d[∫an = ++++ …+

∴an+ 1=++…+++，

∴an+ 1-an=+-=-.]

7.4,7, 10, 15

8. 10

Analysis ∵ =, ∴ n (n+2) = 10× 12, ∴ n = 10.

9.an=2n+ 1

Analytic A 1 = 3, A2 = 3+2 = 5, A3 = 3+2+2 = 7, A4 = 3+2+2 = 9, …, ∴ An = 2n+ 1.

10.55

The number of analytic triangles is: 1, 3,6, 10, 15, …, and the number of the 10th triangle is:1+2+3+4+…+10.

1 1. Solving the problem (1) can be expressed by (-1) n or (-1) n+ 1. The arrangement rule of the absolute value of each term is: the absolute value of the following number is always greater than the absolute value of the previous number by 6, so the general term formula is

(2) The deformation of the sequence is (1-0. 1), (1-0.0 1),

( 1-0.00 1),…,∴an=(n∈n+).

(3) The denominator of each item is 2 1, 22, 23, 24 ... It is easy to see that the numerator of items 2, 3 and 4 is smaller than the denominator. Therefore, the item 1 is changed to-,so the original series can be changed to-,-,...

∴an=(- 1)n (n∈N+)。

(4) unifying the sequence as,,, … for molecules 3, 5, 7, 9, …, which is twice the sum of the serial numbers 1, we can get the general formula of molecules as BN = 2n+ 1, and for denominators 2, 5, 10,1.

Its general formula is an = (n ∈ n+).

(5) an = or an = (n ∈ n+) or an = (n ∈ n+).

12.( 1) Decomposition F (n) = =.

Let n = 10, and the term a 10 = f (10) =.

(2) Solve order = to get 9n = 300.

This equation has no positive integer solution, so it is not an item in this series.

(3) prove that ∵ an = = 1-,

And n ∈ n+,∴ 0

∴0<； An< 1.

∴ All items in the sequence are in the interval of (0, 1).

(4) release the order

Then,

Namely. ∴< n & lt； .

And ∵n∈N+ ∈ n+,∴ If and only if n = 2, the above formula holds, so the series on the interval has terms, and only one term is a2 =.

13 . an =+(- 1)n+ 1

Analytic A =+, B =-,

Therefore, an =+(- 1) n+ 1.

14. The graph (1) has only 1 points and no branches; Figure (2) has two branches except the middle 1 point, and each branch has 1 point; Figure (3) has three branches except the middle 1 point, and each branch has two points; Figure (4) has four branches except the middle 1 point, and each branch has three points; …; Guess that the nth graph has n branches except the midpoint, and each branch has (n- 1) points, then the number of points of the nth graph is1+n (n-1) = N2-n+1.

Functional characteristics of 1.2 sequence

Knowledge carding

2. Positive integer set n+ function value 3. The second item an+1>; One; one

Item 2 an+ 1

work design

1.A2.B3.B

4.c [a 1a2a3 = 32, a 1a2 = 22, a 1a2a3a4a5 = 52, a 1a2a3a4 = 42, then a3 = =, a5 = =.

So a3+a5 =. ]

5.C [a2 =, A3 =, A4 =, so the sequence {an} is a periodic sequence with a period of 3, and it is known that 20 10 can be divisible by 3, so a2.

0 10=a3=。 ]

6.c[∫an = =+ 1

Point (n, an) is on the image of function y =+ 1

Make an image of function y =+ 1 in rectangular coordinate system,

It is easy to know from the image.

When x∑(0), the function monotonically decreases. ∴ A9

When x∑ (,+∞), the function monotonically decreases, ∴ a10 >; a 1 1 & gt； …& gt； a30 & gt 1.

Therefore, among the first 30 items in the sequence {an}, the largest item is a 10 and the smallest item is a9. ]

7.3 2 1-n

8. 12

9.

Analyze ∵ A 1 = 1 and = (n ∈ n+).

∴ ... = ..., that is, an =.

10.-3

Analysis of an ≤ an+ 1 n2+λn≤(n+ 1)2+λ(n+ 1)？ λ≥-(2n+ 1)，n∈N+？ λ≥-3.

1 1.( 1) Prove that an+3 = 1-= 1-

= 1-

= 1-= 1-= 1-

= 1-( 1-an)=an。

∴an+3=an.

(2) The period t = 3 of the sequence {an} known from (1),

a 1=，a2=- 1，a3=2。

∫a 20 10 = A3×670 = A3 = 2，

∴a20 10=2.

12. Solution Because an+1-an = n+1(n+2)-n (n+1) = n+1= n+1,then

When n≤7, n+1>; 0,

When n = 8, n+ 1 = 0,

When n≥9, n+ 1

So a 1

So the term of sequence {an} is the largest, A8 = A9 =.

13.-

Analysis of ∵ an+ 1-an =,

∴a2-a 1=；

a3-a2 =；

a4-a3 =；

… …

an-an- 1 =；

The sum of the above categories is an-a1=++...+

= 1-+-+…+-

= 1-.

∴an+ 1= 1-,∴an=-.

14.

Analysis: (n+1) a-na+Anan+1= 0,

∴[(n+ 1)an+ 1-nan](an+ 1+an)= 0，

∵an & gt； 0,∴an+an+ 1>； 0,

∴(n+ 1)an+ 1-nan=0.

Method 1 =.

∴ … = … ,

∴=.

∫a 1 = 1, ∴ an = A 1 =.

Method 2 (n+ 1) an+ 1-nan = 0,

∴nan=(n- 1)an- 1=…= 1×a 1= 1，

∴nan= 1,an=.