What is the integral part of 1 / 2 + 1 / 3 + 1 / 4 + 1 / 5 + 1 / 6 + 1 / 7 + 1 / 8 + 1 / 9 + 1 / 10 + 1 / 11 + 1 / 12? Why

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• 1. Observe the equations in the following order: 9 × 0 + 3 = 3, 9 × 1 + 4 = 13 Guess: the nth equation (n is a positive integer) should be expressed as?
• 2. Observe the following equation: 9 × 0 + 1 = 19 × 1 + 2 = 119 × 2 + 3 = 219 × 3 + 4 = 319 × 4 + 5 = 41 According to the law reflected in the number table, we guess that the nth equation (n is a positive integer) should be () A. 9（n-1）+n=10（n-1）+1B. 9n+n=（n-1）+1C. 9n+（n-1）=n2-1D. 9n+n+1=10n+1
• 3. How to easily calculate the square of 1.2222 multiplied by 9-1.3333 multiplied by 4?
• 4. Calculate the square division of - 1 by 1 / 9 times (- 2)
• 5. As shown in Figure 1, ab ∥ CD, try to guess the relationship between ∠ bed and ∠ B, ∠ D? Please explain the reason
• 6. As shown in the figure, given ab ∥ CD, guess what is the relationship between ∠ B, ∠ bed, ∠ D in Figure 1, figure 2 and figure 3? Please show their relationship by equation and prove one of them. (1)______ ；（2）______ ；（3）______ ．
• 7. Observe the following set of equations 1 + 3 = 2, square 1 + 3 + 5 = 3, square 1 + 3 + 5 + 7 = 4, according to this rule - 1-3-5-7. [2N-1] where n is positive Calculate the result
• 8. The second power of 3 - the second power of 1 = 8 = 8X1, the square of 5 - the square of 3 = 16 = 8x2?
• 9. What does it mean that my parents' birthdays add up to my birthday? Mom 0815 + dad 0216 = me 1031 What's the point of coincidence?
• 10. What is the theorem / conjecture that the cube of any natural number a is equal to the sum of a continuous odd numbers? I want to know its name Has any textbook mentioned this theorem If yes, please tell me the publishing house, grade and page number
• 11. The first line is 1, the second line is 2, 3, the third line is 4, 5, 6, 7, and the fourth line is 8.9.10.11.12.13.14.15
• 12. What holds in the following equation is the square of () a.a-2a-1 = (A-1) and the square of B.A + A + 1 / 4 = (A-1 / 2) C. Square of B - 6b-9 = (B-3) square C. Square of M + 1 / 6m + 1 / 144 = (M + 1 / 12)
• 13. In the formula (a + 1) ^ 2 = a ^ 2 + 2A + 1, when a takes 1, 2, 3 respectively N, the following equation can be obtained: (1 + 1) ^ 2 = 1 ^ 2 + 2 * 1 + 1 (2+1)^2=2^2+2*2+1 (3+1)^2=3^2+2*3+1 …… (n+1)^2=n^2+2*n+1 Using the above formulas, conjecture and prove the summation formula, 1 + 2 + 3 + 4 + +n=____ (expressed by the algebraic expression of n)
• 14. 14. In the formula (a + 1) 2 = A2 + 2A + 1, when a takes 1,2,3 , N, we can get the following n equations (1 + 1) 2 = 12 + 2 × 1 + 1 （1+1）2=12+2×1+1 （2+1）2=22+2×2+1 （3+1）2=32+2×3+1 …… （n+1）2=n2+2×n +1 By adding the left and right sides of these n equations, the summation formula can be derived 1+2+3…… +N = (expressed in an algebraic expression containing n)
• 15. 1 1 1 1 ——+——+——+...+—————= 2X4 4X6 6X8 2008X2010 1/2X4 + 1/4X6 + 1/6X8 + ...+ 1/2008X2010
• 16. There is a class of numbers, 9-1 = 8,16-4 = 12,25-9 = 16,36-16 = 20. Q n is a positive integer, which is expressed as -?
• 17. Observe the following equations: 9-1 = 8,16-4 = 12,25-9 = 16,36-16-20, ', try the equation about n to express the law you found
• 18. Observe the following equations: 9-1 = 8, 16-4 = 12, 25-9 = 16, 36-16 = 20 Let n be a positive integer. The following equation is () A. （n+2）2-n2=4（n+1）B. （n+1）2-（n-1）2=4nC. （n+2）2-n2=4n+1D. （n+2）2-n2=2（n+1）
• 19. 9-1=8;16-4=12;25-9=16;36-16=20；… What laws do these equations reflect among positive integers
• 20. Observation formula 9-1 = 8; 16-4 = 12; 25-9 = 16; 36-16 = 20 These equations reflect some rules between positive integers, If n is a positive integer, the law can be expressed by the formula of n