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Gauss arithmetic: 1 + 3 + 5 + 7 +. + 2003 =, 1 + 4 + 7 + 10 + 13 +. + {3n-2}
(first item + last item) * number of items / 2
(1+2003)*(2003/2)/2=1003503
(1+3n-2)*((3n-2)/3)/2
In △ ABC, a, B and C are the opposite sides of angles a, B and C, and B2 = AC, then the value range of ∠ B is______ ．
In △ ABC, substituting B2 = AC into cosine theorem, we obtain A2 + c2-2ac · CoSb = B2 = AC, ∩ CoSb = A2 + C2 − ac2ac ≥ 2Ac − ac2ac = 12, ∩ 0 < B ≤ π 3, so the answer is: (0, π 3]
In △ ABC, a = x, B = 2, B = 45 °, if the triangle has two solutions, then the value range of X is ()
A. x＞2B. x＜2C. 2＜x＜22D. 2＜C＜23
If a ≤ 45 °, then C ≥ 90 ° so that a + b > 180 ° does not hold. If a = 90, then the complementary angle is also 90 °, so 22 < Sina < 1A = 22sina, so 2 < a < 22, so C is selected
RELATED INFORMATIONS
- 1. In △ ABC, a = xcm, B = 2cm, B = 45 ° are known. If there are two solutions of triangle by using sine definite, then the value range of X is () A. 2＜x＜22B. 2＜x≤22C. x＞2D. x＜2
- 2. In the known triangle ABC, BC = x, AC = 2, B = 45 degrees, if the triangle has two solutions, then the value range of X is? (2,2√2) Draw an angle of 45 ° with B as the vertex and BC = x on one side Take point C as the center of the circle, radius 2, draw a circle, because there are two solutions, so the circle should intersect with the other side, if there is no solution, it is separated, if there is a solution, it is tangent After you draw the picture, you can clearly see that the distance from point C to the other side (the distance from the center of the circle to the chord), that is, the height of the triangle ABC is less than the radius, that is, xcos 45 ° < 2 In addition, BC edge is larger than radius, that is, x > 2 two
- 3. In triangle ABC, a = 2, B = 1, find the value range of angle B
- 4. In the sequence {an}, if the first n terms and Sn = 3n-2n ^ 2 (n belongs to n *), then an =? Second, what is the size relationship between Sn and Nan?
- 5. Let x ^ 2n = 3, find (1 / 3x ^ 3n) · [4 (x ^ 5) ^ n] Let x ^ 2n = 3, find (1 / 3x ^ 3n) ^ 3 · [4 (x ^ 5) ^ n]
- 6. It is stipulated that the "H operation" of positive integer n is: ① when n is odd, H = 3N + 13; ② when n is even, H = NX1 / 2x1 / 2 Where h is odd Q 7 after 2010 times of "H operation", how much is it equal to?
- 7. How many squares are there in 1-200 natural numbers
- 8. If M and N are natural numbers and M is a multiple of N, then what is the maximum divisor of M and N?
- 9. M is a natural number greater than 10, M + 2 is a multiple of 3, M + 3 is a multiple of 4, M + 4 is a multiple of 5, M + 5 is a multiple of 6, what is m?
- 10. The fractional unit of n of M (both M and N are non-zero natural numbers) is (). He has () such units
- 11. If the function f (x) = (m-2) x multiplied by X + (m-1) x + 2 is even, find the monotone decreasing interval of F (x + 1) and explain the reason
- 12. The function f (x) whose domain is n + and whose value is also a positive integer: for any n ∈ n +, f (n + 1) > F (n); f (f (n)) = 3N, find f (4), f (5) We know that {f (n)} is a strictly increasing positive integer sequence -- > F (n) ≥ n f(f(1))=3≤f(3)--->f(1)≤3 If f (1) = 1, it is contradictory to f (1)) = 3; if f (1) = 3 --- > F (1)) = f (3) = 3, it is contradictory So: -- > F (1) = 2, f (2) = f (f (1)) = f (2) = 3 f(3)=f(f(2))=6 f(6)=f(f(3))=9 Because {f (n)} is a strictly increasing positive integer sequence -->f(4)=7,f(5)=8 How did you get this step? f(6)=f(f(3))=9 Because {f (n)} is a strictly increasing positive integer sequence
- 13. When n is a positive integer, the function n & nbsp; (n) is defined to represent the largest odd factor of N. for example, n & nbsp; (3) = 3, n & nbsp; (10) = 5 Let s (n) = n (1) + n (2) + n (3) + +Then (1) s (4) = n (2n)___ ．（2）S（n）= ___ ．
- 14. Given that f (x) satisfies f (x + y ^ 2) = f (x) + 2 (f (y)) ^ 2 for any natural number x and y, and f (1) is not equal to 0, then f (2012) =?
- 15. The sum of the first n items of the arithmetic sequence is a, the sum from the N + 1 to the 2n items is B, and the sum from the 2n + 1 to the 3N items is C. (1) the known tolerance is D, and D is b-a (2) Prove that a, B and C are equal difference sequence
- 16. The ratio of the sum of the first n terms of two arithmetic sequences is 2n / (3N + 1), and the ratio of the nth term of two arithmetic sequences is calculated
- 17. It is known that the sequence {an} is an arithmetic sequence, A1 = 2, a7 = 11. If five numbers are inserted between every two terms, they will form a new arithmetic sequence with the number of the original sequence What is the 65th term of the new sequence?
- 18. It is known that {an} is an arithmetic sequence, A1 = 2, A2 = 3. If three numbers are inserted between every two adjacent terms to form a new arithmetic sequence with the number of the original sequence, we can find: (1) what is the 12th term of the original sequence? (2) What is the 29th item of the new sequence?
- 19. If the sum of the first n terms of an arithmetic sequence is 48 and the sum of the first 2n terms is 60, then the sum of the first 3N terms of an arithmetic sequence is 48 Is it 72?
- 20. The sum of the first n terms of the arithmetic sequence is 30, and the sum of the first 2n terms is 100?