Mathematics Exercises of Grade Five in China (III) 1 arithmetic progression Sum. Starting from the second number, the difference between each number and its previous number is a fixed number. This series is called arithmetic progression, and this constant is called tolerance. For example: (1) 1, 2, 3, 4, 5, ... 99,100 (2)1,3, 5, 7, 9, ... 97, 99 (3) 4. If the number of projects in a series is limited, we call the first project the first project and the last project the last project. Sum of arithmetic progression = (first term+last term) × number of terms ÷2 Last term = first term+tolerance× (number of terms-1) First term = last term-tolerance× (number of terms-1) Number of terms = (last term-first term) \. Example 2 Find the sum of arithmetic progression whose first term is 5, the last term is 155 and the number of terms is 5 1. There are 60 numbers in Example 3. The first number is 7. Starting with the second number in the series 3, 8, 13, 18, ... In Example 4, how much is the last number more than the previous 80th item? If a number is greater than 4, find the sum of these 60 numbers. Example 5 3+7+ 1 1+...+99 =? Example 6 A arithmetic progression whose term is 15, and the last term is 1 10 with a tolerance of 7. What's the sum of this arithmetic progression? Five years in profit and loss problem (3) 1, a tree planting team went to plant trees. If each person plants 5 trees, there are still 14 seedlings left; If each race has seven trees, there will be a shortage of four seedlings. How many people are there in this group? A * *, how many seedlings? The school bought some basketballs and distributed them to all classes equally. If each class is divided into four, it is14; If each class is divided into five, it's just over. How many basketballs did the school buy? How many classes are there? The kindergarten class in Yan Xi Street distributes apples to the children. If everyone is divided into six parts, 72 parts are missing; If everyone is divided into four parts, it's just over. Find out the number of children in this kindergarten class and the total number of apples distributed. 4, a workshop production plan, scheduled to produce a number of parts. If each group completes 16 pieces, it can exceed 6 pieces; If each group completes 15 pieces, it can be more than 2 pieces. How many parts does this workshop plan to produce? How many groups of workers are there? 5. Fourth grade 1 class, reward excellent students with pencils. If everyone wins 14, there will be19; If everyone wins 12, there will be 1 1. How many excellent students are there in this class? How many pencils are there? Xiaohua leaves home for school at 7 o'clock every morning. Walk 60 meters per minute, 6 points late; If you walk 80 meters per minute, you can get to school three minutes earlier. How many minutes does it take to get to school from home on time? How far is Xiaohua's home from school? 7. Measure the height of the bridge with a rope. When the rope is folded in half, there is still 5 meters left when it is hoisted to the water surface, and when it is folded in three, there is still 2 meters left when it is hoisted to the water surface. Find the height of the bridge and the length of the rope. In the math contest calculated according to the new definition in the fifth grade exercise (4), there is a question that needs to be calculated according to the new definition. The characteristic of this kind of problem is that it specifies a new definition of operation symbols and a new operation order, and requires a new operation method according to the new definition. The questions calculated according to the new definition are both interesting and flexible. Although it is different from the mathematics knowledge in the textbook, we can answer it with what we have learned. The key to solve this problem is to correctly understand the definition and transform the problem into four well-known operations according to the newly defined relationship. Answering such questions helps to improve our observation ability, analysis ability, adaptability and calculation ability. Example 1 2 3 = 2+22+222 = 246, 3 4 = 3+33+333 = 3702, ... According to this law: (1) 32; (2)5 3; (3) 1 X= 123, find X. Example 2 shows that a ※ b = (a+b) × (a-b), and Example 3 stipulates that 1※4= 1×2×3×4, find 20. 6※5=6×7×8×9× 10, then (4※5)÷(6※3)=? Example 4 stipulates that [a, b, c, d] = 9ab-cd, and Example 5 assumes that a*b represents 4 times of A minus B. If [1, 2,3, X]=3, the value of X is 3 times, that is, a * b = 4a-3b. (1) calculation: (1.5 * 0.8) * 0.5; (2) Given X*(5*2)=46, find X. Example 6 If a > b, then [A, b] = a; If a < b, then [A, b] = B. Try (1) [8,0.8]; (2){[ 1.9, 1.90], 1.9} Example 7 n is a natural number, and it is specified that f (n) = 3n-2, for example, f (3) = 3× 3-2 = 7. Try to find the value of f (1)+f (2)+f (3) ... +f (100). Example 8 If 1= 1! 1×2=2! 1×2×3=3! …… 1×2×3……× 100= 100! Then 1! +2! +3! +……+ 100! The unit number of is (). The problem (4) in the math classroom exercise of grade five in China is reduced to the problem 1, and there is a number. Multiply it by 5 and subtract 26, then divide the difference by 4 and add 13, and finally get 29. What is the number? 2. The workshop will issue bonuses according to the workers' overcapacity. Give half of the total bonus to Party A, then give the remaining half to Party B, then to Party C, 80 yuan, Ding 7 yuan and finally to 4 yuan. How much is this bonus? 3. An old man said, "Add my age to 17, then divide it by 4, subtract 15, and multiply it by 10, which is exactly 100." How old is this old man? 4. there are two numbers, a and b, and the result of subtracting b from a is equal to 7; B plus a, then multiply a, then subtract a, and finally divide by a, the result is equal to a ... find the numbers a and B. A peach seller took a basket of peaches to various stores to sell: in the first store, he tasted one first, and then bought the remaining half; In the second restaurant, I tasted one first and then bought the remaining half. When you get to the third place, try one first and buy the remaining half. At this time there are 35 peaches left in the basket. How many peaches are there in this basket? 6. Someone went out for a trip, and spent more than half of the 350 yuan on the trip. When he came back, he spent the remaining half, which was 130 less, leaving 285 at home. How much money did he bring with him on this trip? 7. Dongxing Machinery Factory has five workshops. It is planned to produce twice as many lathes this year as last year, resulting in 480 more lathes than planned. It is known that even if each workshop produces less 120 units, it can reach 800 units. How many lathes did this factory produce last year? 8. Add 1 to a certain number, subtract 2, multiply 3 and divide 4, and the result is 6. What is the number? Fifth grade exercise (5) Count a Pentagon and connect its diagonals into pentagons (as shown on the right). How many triangles are there in a * * * in the picture? Problems like this are the counting problems of graphs. When counting, it is required not to repeat or omit. Example 1 How many line segments are there in the picture below? Example 2 How many line segments are there in the picture on the right? A B C D E Example 3 How many triangles are there in the right * * *? Example 4 How many triangles are there in the regular Pentagon shown below? A E B D C Example 5 Count the total number of squares in the picture below (). Example 6 Statistics There are () rectangles in the figure below.