What rules do you find by observing the following formulas 1^2=(1×2×3)/6 1^2+2^2=(2×3×5)/6 1^2+2^2+3^2=(3×4×5)/6 1^2+2^2+3^2+4^2=(4×5×9)/6 … (1) Can you express this rule with a formula? (2) According to the rule you found, calculate 1 ^ 2 + 2 ^ 2 + 3 ^ 2 + +8^2 Second, the process

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• 1. If you look at the following formula, you will find the rule The second power of 1-0 = 1; the second power of 2-1 = 3; the second power of 3-2 = 5; the second power of 4-3 = 7; the second power of 5-4 = 9
• 2. What rules can you find by observing the following formulas? 1*3+1=4=2^2 2*4+1=9=3^2 3*5+1=25=5^2. (1) Please write two more equations according to your observation (2) According to the same number equation, please put the rules you find out in letters
• 3. Please observe the following formulas carefully, and then write the fourth and fifth formulas according to the rules you found √16=√1x16=√1x4²=√1x√4²=1x4=42.√32=√2x16=√2x4²=√2x√4²=√2x4=4√23.√48=√3x16=√3x4²=√3x√4²=√3x4=4√34.√64=__________________ 5.√80=_____________________ Please write the nth equation that agrees with the above rule. (n ≥ 1, and it is an integer)
• 4. Mathematics (4th power of a) + 6th power of a * 2nd power of a
• 5. The second power of 152 - the second power of 52 divided by the second power of 284 - the second power of 16 is in the form of fraction
• 6. Use the eight numbers 2, 3, 4, 5, 6, 7, 8 and 9 to form two four digit numbers. How to make the product of the two numbers maximum? Pay attention to the analysis process. OK, add money
• 7. Use 2378 to form two two digit numbers to make their product maximum. What are the two two digit numbers? What is the maximum value?
• 8. The ten numbers of 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are composed of two 4-digit and one 2-digit Make two digits plus four digits equal to another four digits. More than ten digits must be used and can only be used once. Try to write ten different expressions
• 9. Take three different numbers from 1,3,5,7,9, and then take two different numbers from 0,2,4,6,8, to form five digits larger than 70123. How many are there
• 10. Select nine numbers from the ten numbers of 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 to form a two digit number, a three digit number and a four digit number, so that the sum of the three numbers equals 2010, then the unselected number is______ ．
• 11. Observe the following formulas: 31 = 3 & nbsp; & nbsp; & nbsp; 32 = 9 & nbsp; & nbsp; & nbsp; & nbsp; 33 = 27 & nbsp; & nbsp; & nbsp; & nbsp; 34 = 81 & nbsp; & nbsp; & nbsp; & nbsp; 35 = 243 & nbsp; & nbsp; & nbsp; & nbsp; 36 = 729 & nbsp; & nbsp; & nbsp; 37 = 2187 & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; 38 = 6561 Use the rule you found to judge the last number of 32004 is () A. 3B. 9C. 7D. 1
• 12. Looking at the following formulas, what rules do you find? ①16×14=224=1×（1+1）×100+6×4 ②23×27=621=2×（2+1）×100+3×7 ③32×38=1216=3×（3+1）×100+2×8 Simple description of the above found rules: mainly text and letters
• 13. Observe the following formula and fill in the blanks 1.07÷9＝0.118 1111.04÷9＝﹙ ﹚ 11.06÷91.228 ﹙ ﹚÷9＝﹙ ﹚ 111.05÷9＝12.338 ﹙ ﹚÷9=﹙ ﹚
• 14. First observe the following formula, fill in the blanks and summarize the rules 3²-1²=8×1, 5²-3²=8×2； Then (1) 7 & sup2; - 5 & sup2; = 8 × (3) （2）9²-(7)²=8×4 （3）(11)²-9²=8×5 Sum up the law____________________________________ .
• 15. 2 + 4 = 2 × 3 & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; 2 + 4 + 6 = 3 × 42 + 4 + 6 + 8 = 4 × 5 & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; 2 + 4 + 6 + 8 + 10 + +100=______ ×______ ．
• 16. Observe each of the following formulas and fill in the blanks according to the rules you find 4*5=20,3*6=18 5*6=30,4*7=28 6*7=42,5*8=40 . If 1222 * 1223 = 1494506, then 1221 * 1224 = () Write the relationship
• 17. Observe the following three groups of formulas, find the rules and fill in the blanks 1 × 2 = 2 × 3 of 3, 1 × 2 + 2 × 3 = 2 × 3 × 4 of 3, 1 × 2 + 2 × 3 + 3 × 4 = 3 × 4 × 5 of 3 (1) 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 = 4 × 5 × () (2) 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + 5 × 6 = () × () × () （3）1×2+2×3+3×4+… +N × (n + 1) = () × () × () (4) 1 × (1 × 2 + 2 × 3 + 3 × 4 +) + 199 × 200) = (this one needs to be written)
• 18. Carefully observe the rules of the following group of formulas and write two such formulas: 99-27 = 72, 88-35 = 53, 77-25 = 52, 55-23 = 32
• 19. 32-12=8×1；52-32=8×2；72-52=8×3… Observe the above formula, what law can you find? Please use algebra to express, and use this law to calculate the value of 20012-19992
• 20. Observe the following equations, find the laws from them, and use them to find the values of the following equations Look at the following equations: 1/√2+1=√2-1,1/√3-√2,1/√4-√3,1/√5+√4=√5-√4,… Find the laws from the above equations and use them to find the values of the following equations 1/√2+1+1/√3+√2+1/√4+√3+…… +1/√2013+√2012