Summary of knowledge points of arithmetic mean value in senior one.
The general formula of arithmetic progression is: an = a1+(n-1) d.
Or an=am+(n-m)d
The first n terms and formulas are: sn = na1+[n (n-1)/2] d or sn=(a 1+an)n/2.
If m+n=2p, then: am+an=2ap.
All the above n are positive integers.
Text translation
Value of material n = first material+(material number-1)* tolerance.
Sum of the first n items = (first item+last item) * Number of items /2
Tolerance = Last Item-First Item
Exercises and Analysis of Arithmetic Average Item in Senior One Mathematics
1. If the first term of arithmetic progression {an} a 1= 1 is known and the tolerance d=2, then a4 is equal to ().
a5b . 6
C.7 D.9
Answer: c
2. In the sequence {an}, if a 1= 1, an+ 1=an+2(n? 1), then the general term formula of the series an= ()
A.2n+ 1 B.2n- 1
C.2n D.2(n- 1)
Answer: b
3. Delta ABC three internal angles A, B and C form a arithmetic progression, then B = _ _ _ _ _ _ _ _
Analysis: ∫A, B and C are arithmetic progression. 2B=A+C
A+B+C= 180？ ,? 3B= 180？ ,? B=60？ .
Answer: 60?
4. In arithmetic progression {an},
(1) a5=- 1 and a8=2 are known. Find a 1 and d;
(2) Given a 1+a6= 12 and a4=7, find a9.
Solution: (1) a 1+ From the meaning of the question? 5- 1? d=- 1，a 1+？ 8- 1? d=2。
The solution is a 1=-5 and d= 1.
(2) a 1+a 1+ From the meaning of the question? 6- 1? d= 12，a 1+？ 4- 1? d=7。
The solution is a 1= 1, and d=2.
? a9 = a 1+(9- 1)d = 1+8？ 2= 17.
First, multiple choice questions
1. In arithmetic progression {an}, if a 1=2 1 and a7= 18, the tolerance d= ().
12
C.- 12 D.- 13
Analysis: choose C. ∫ A7 = a1+(7-1) d = 21+6d =18. d=- 12。
2. In arithmetic progression {an}, a2=5 and a6= 17, then a 14= ().
A.45 B.4 1
C.39 D.37
Analysis: Choose B.a6=a2+(6-2)d=5+4d= 17, and get d=3. So a14 = a2+(14-2) d = 5+12? 3=4 1.
3. The known sequence {an} pairs any n? N* and the point Pn(n, an) are both on the straight line y=2x+ 1, then {an} is ().
A. arithmetic progression with an error of 2 B. Arithmetic progression with an error of 1
C. arithmetic series with an error of-2 d. Non-arithmetic series
Analysis: choose A.an=2n+ 1,? An+ 1-an=2, you should choose a.
4. Given that the arithmetic mean of m and 2n is 4, 4, and the arithmetic mean of 2m and n is 5, the arithmetic mean of m and n is ().
A.2 B.3
C.6 D.9
Analysis: choose b, from the meaning of the question, m+2n=82m+n= 10. m+n=6，
? The arithmetic mean of m and n is 3.
5. In the following series, arithmetic progression has ().
①4,5,6,7,8,? ②3,0,-3,0,-6,? ③0,0,0,0,?
④ 1 10,2 10,3 10,4 10,?
A. 1
C.3 D.4
Analysis: Choose C. Use arithmetic progression's definition to verify that ①, ③ and ④ are arithmetic progression.
6. The series {an} is a arithmetic progression with a first term of 2 and a tolerance of 3, and the series {bn} is a arithmetic progression with a first term of -2 and a tolerance of 4. If an=bn, the value of n is ().
A.4 B.5
C.6 D.7
Analysis: choose B.an=2+(n- 1)? 3=3n- 1，
bn=-2+(n- 1)？ 4=4n-6，
Let an=bn get 3n- 1=4n-6,? n=5。
Second, fill in the blanks
7. Given arithmetic progression {an} and an=4n-3, the first term a 1 is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
Analysis: a 1=4 comes from an=4n-3? 1-3= 1，d=a2-a 1=(4？ 2-3)- 1=4, so the first term a 1 of arithmetic progression {an} is 1, and the tolerance d=4.
Answer: 1 4
8. In arithmetic progression {an}, a3=7, a5=a2+6, then A6 = _ _ _ _ _ _ _
Analysis: Let the tolerance of arithmetic progression be D and the first term be a 1, then A3 = A1+2D = 7; a5-a2=3d=6。 ? d=2，a 1=3。 ? a6=a 1+5d= 13。
Answer: 13
9. It is known that the sequence {an} satisfies a2n+ 1=a2n+4, and a 1= 1, an>0, then an = _ _ _ _ _ _
Analysis: According to the known condition a2n+ 1=a2n+4, that is, a2n+ 1-a2n=4,
? The sequence {a2n} is an arithmetic series with a tolerance of 4,
? a2n=a2 1+(n- 1)？ 4=4n-3。
∵an & gt； 0,? an=4n-3。
Answer: 4n-3
Third, answer questions.
10. In arithmetic progression {an}, it is known that a5= 10, a 12=3 1. Find its general formula.
Solution: obtained by an = a1+(n-1) d.
10 = a1+4d31= a1+1d, and the solution is a 1=-2d=3.
? The general formula of arithmetic progression is an=3n-5.
1 1. It is known in arithmetic progression {an}, a 1.
(1) Find the general term formula {an }； of this series;
(2) Is 268 a project in this series? If yes, what is the quantity? If not, explain why.
Solution: (1) is given by a3=2 and a6=8.
And ∵{an} is arithmetic progression, let the first term be a 1 and the tolerance be d,
? A 1+2d=2a 1+5d=8, the solution is a 1=-2d=2.
? an=-2+(n- 1)？ 2
=2n-4(n？ N*)。
? The general formula of the sequence {an} is an=2n-4.
(2) Let 268=2n-4(n? N*), the solution is n= 136.
? 268 is item 136 in this series.
12. It is known that (1, 1) and (3,5) are two points on arithmetic progression's {an} image.
(1) Find the general term formula of this series;
(2) draw this series of images;
(3) Judge the monotonicity of this series.
Solution: (1) Because (1, 1) and (3,5) are two points on the arithmetic progression {an} image, a 1= 1, a3=5, because a3 = a/kloc-5.
(2) The image is some equidistant points on the straight line y=2x- 1 (as shown in the figure).
(3) Because the linear function y=2x- 1 is increasing function,
So the sequence {an} is an increasing sequence.