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Observe the deformation law of the following equation: 2 / (1 × 3) = 1 - (1 / 3), 2 / (2 × 4) = 1 / 2-1 / 4, 2 / 3 × 5 = 1 / 3-1 / 5,... Please answer the following questions; (1) If n is a positive integer, guess 2 / N (n + 2)= (1) Explain the reason for the conjecture (3) Sum: 1 / 1 × 3 + 1 / 2 × 4 + 1 / 3 × 5 +... + 1 / 49 × 51
(1) 2 / N (n + 2) = 1 / n-1 / (n + 2) (2) 1 / n-1 / (n + 2) = (n + 2-N) / N (n + 2) = 2 / N (n + 2) (3) let the sum be s, then 2S = [2 / (1 * 3) + 2 / (3 * 5) +... + 2 / (49 * 51)] + [2 / (2 * 4) + 2 / (4 * 6) +... + 2 / (48 * 50) = (1-1 / 3 + 1 / 3-1 / 5 +... + 1 / 49-1 / 51) + (1 / 2-1 / 4 + 1 / 4-1 / 6 +... + 1 / 48-1 /
RELATED INFORMATIONS
- 1. The following deformation rules are observed 1／(1*2)=1-1/2; 1/(2*3)=1/2-1/3; 1/(3*4)=1/3-1/4;…… . sum: 1 / (1 * 2) + 1 / (2 * 3) + 1 / (3 * 4) + ..+1/（2009*2010）
- 2. Observe the following formula: 1 ^ 2 + (1 * 2) ^ 2 + 2 ^ 2 = (1 * 2 + 1) ^ 2 ^ 2 + (2 * 3) ^ 2 + 3 ^ 2 = (3 * 2 + 1) ^ 2 ^ 2 + (3 * 4) ^ 2 + 4 ^ 2 = (3 * 4 + 1) ^ 2 Observe the following rules 1^2+(1*2)^2+2^2=(1*2+1)^2 2^2+(2*3)^2+3^2=(3*2+1)^2 3^2+(3*4)^2+4^2=(3*4+1)^2 ① Write the 2011 formula ② Write the formula on line N and prove your conclusion
- 3. S = 1 + 2 ^ - 1 + 2 ^ - 2 + 2 ^ - 3 + +2 ^ - 2010, please calculate the formula on the right to get the value of S, (hint: multiply by 2 on both sides of the equation) do you see the law?
- 4. When a = 2 and B = 3, calculate the values of (a + b) 2 - (a-b) 2 and 4AB respectively, and find out what law (expressed by the formula containing a and b)
- 5. Specify a kind of operation: a * b = (a + 5 / 2) × (B-3 / 4) △ B, try to calculate 1 / 2 * 5 / 4
- 6. 1 / 2 = 1-1 / 2, 1 / 6 = 1 / 2-1 / 3, 1 / 12 = 1 / 3-1 / 4
- 7. 1 / 2 = 1-2 / 1, 1 / 6 = 1 / 2-3 / 1, 1 / 12 = 1 / 3-4 / 1... According to the above rules, calculate the following question: 1 / 2 + 1 / 6 +1 in 12 + 1 in 20 + 1 in 30
- 8. 1 / 2 = 1-1 / 2, 1 / 6 = 1 / 2-1 / 3, 1 / 12 = 1 / 3-1 / 4. According to the above rule, calculate the following questions: 1 / 2 + 1 / 6 + 1 / 12 + 1 / 20 + 1 / 30
- 9. It is known that 5 / 6 = 1 / 2 + 1 / 3, 7 / 12 = 1 / 3 + 1 / 4, 9 / 20 = + 1 / 4 + 1 / 5 1 + 2 1-6 5 + 12 7-20 9 + 30 11-42 13 Let's say what we mean! (yes, there is a reward)
- 10. A formula for finding the fraction of one number to another The machinery factory produced a batch of machines, 20 in the first day, and 25% more in the second day than in the first day. The number of machines produced in the two days is exactly the same as that of this batch of machines 37.5%. How many machines are there in this batch?
- 11. Observe the following deformation law: 1 / (1 × 2) = 1-1 / 2; 1 / (2 × 3) = 1 / 2-1 / 3; 1 / (3 × 4) = 1 / 3-1 / 4 seek 1/(1×2)+1/(2×3)+1/(3×4）+…… +1/(2011×2012)·
- 12. The beginning of a composition about mathematics
- 13. How to write a math composition?
- 14. All the answers to the Yangtze River exercise book
- 15. Simple equation in mathematics book published by people's Education Press ·Is the horizontal line of (2) in example 4 6 A or 6 × a I asked for leave to study this
- 16. Fifth grade mathematics volume 1 exercise book equation application problem (1) A primary school organized students to visit the botanical garden. A total of 840 people visited the botanical garden throughout the day. The number of people visiting the botanical garden in the afternoon was twice that in the morning. How many people visited the botanical garden in the morning and afternoon? (2) Xiao Ming said: Dad is three times my age this year. Dad said: I am 22 years older than Xiao Ming. How old are dad and Xiao Ming this year? (3) What is the basis of solving the equation? What should we pay attention to? (4) Xiaoming said: we have 50 books in total. Xiaoming said: give me five of your books, and we have the same number of books. How many books do you have?
- 17. Fill in the appropriate relevant words 1. Our great motherland, vast territory and abundant resources, has something to be proud of in history and culture 2. () it's raining cats and dogs, () the fire fighters are still training in the rain
- 18. Mathematics problem of grade 6 (1) Master Wang weaves two fifths of a meter long blanket in two-thirds of an hour. How many meters does he weave in one hour? (2) Master Li weaves two fifths of a meter long blanket every hour. How many meters does he weave in two-thirds of an hour? (3) Master Zhang weaves two fifths of a meter long blanket every hour. How many hours does it take to weave two-thirds of a meter long blanket?
- 19. First, fill in the quantitative relationship of each question and solve the following equation: The speed of an airplane is 452 km / h, and that of a motorcycle is x km / h. The speed of an airplane is 18 times that of a motorcycle () = aircraft speed （ ）
- 20. What should be filled in the brackets of quantity relation questions [12,19,29,47,78127, ()]?