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(1-1/2)+(1/2-1/3)+(1/3-1/4)+···+(1/2009-1/2010
To find the sum of the given series, we need to add all the terms together. (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/2009 - 1/2010) We can simplify each term by finding the common denominator. 1 - 1/2 = 2/2 - 1/2 = 1/2 1/2 - 1/3 = 3/6 - 2/6 = 1/6 1/3 - 1/4 = 4/12 - 3/12 = 1/12 We can observe that each term follows this pattern - the denominator of the second fraction becomes the denominator of the first fraction in the next term. So, the series can be written as: 1/2 + 1/6 + 1/12 + ... + 1/2009 To find the sum of this series, we need to find the common denominator of all the fractions. The common denominator of 2, 6, 12, ..., 2009 will be the least common multiple (LCM) of these numbers. Calculating the LCM of these numbers is a bit lengthy. Instead, we can find the LCM of 2, 3, 4, ..., 2010, and then divide by the LCM of 2, 3, 4, ..., 2009. LCM(2, 3, 4, ..., 2010) / LCM(2, 3, 4, ..., 2009) = 2010 / 2 = 1005 So, the common denominator is 1005. To add the fractions, we need to express them with the common denominator: 1/2 = (1/2) * (1005/1005) = 1005/2010 1/6 = (1/6) * (1005/1005) = 167.5/2010 1/12 = (1/12) * (1005/1005) = 83.75/2010 Now we can add: 1005/2010 + 167.5/2010 + 83.75/2010 + ... + 1/2009 We can observe that the denominators of the fractions form an arithmetic sequence, and the numerators follow the same pattern. Using the formula for the sum of an arithmetic sequence: Sum = (first term + last term) * number of terms / 2 In this case, the first term is 1005/2010, the last term is 1/2009, and the number of terms is 2010. Sum = (1005/2010 + 1/2009) * 2010/2 Sum = (1005/2010 + 1/2009) * 1005 Sum = (1005 * 2009 + 1 * 2010) / 2 Sum = (2019955 + 2010) / 2 Sum = 2021965 / 2 Sum = 1010982.5 Therefore, the sum of the given series is 1010982.5.
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孩子一考试就马虎、粗心大意,哪些方法能够改掉孩子粗心坏习惯?
改掉孩子的粗心坏习惯需要一定的耐心和持续的努力。以下是一些可以改善孩子粗心坏习惯的方法: 1. 制定明确的目标和计划:帮助孩子设立明确的目标,让他们明白为什么精确性和细致性在学习中的重要性,并和他们一起制定一个改善计划。 2. 提供适当的环境:为孩子提供安静、整洁、没有干扰的学习环境,可以让他们更加专注和注意细节。 3. 建立良好的学习习惯:帮助孩子建立每天定期学习的习惯,并鼓励他们养成整理笔记、归档作业等良好的学习习惯。 4. 分解任务:将复杂的任务分解为小的、易于处理的步骤,这样可以让孩子更容易掌握和管理。 5. 使用提醒工具:使用提醒事项、闹钟或手机应用程序,帮助孩子记住重要的事情和任务。 6. 强化奖励机制:设立奖励机制,奖励孩子遵守自己的计划、完成任务和保持专注的行为。奖励可以是一些小的奖品、夸奖或特别的活动。 7. 耐心和示范:给予孩子足够的时间和支持,耐心地引导他们学习如何更加细致和注意细节。同时,以身作则,展示给孩子一个良好的榜样。 8. 找到适合的学习方法:了解孩子的学习习惯和兴趣,帮助他们找到适合自己的学习方法,这样可以增加他们的注意力和自觉性。 需要注意的是,改变习惯需要时间和持续的努力。家长和教师应该形成合作关系,共同关注孩子的学习进展,并给予积极的支持和鼓励。
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