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Misc 32 - 150 workers were engaged to finish a job in - Finding sum from nth number

  1. Chapter 9 Class 11 Sequences and Series
  2. Serial order wise
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Misc 32 150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed. Let total work = 1 and let total work be completed in = n days Work done in 1 day = (𝑇𝑜𝑡𝑎𝑙 𝑤𝑜𝑟𝑘)/(𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑦𝑠 𝑡𝑜 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 𝑤𝑜𝑟𝑘) = 1/𝑛 This is the work done by 150 workers Work done by 1 worker in one day = 1/150𝑛 Given that In this manner it took 8 more days to finish the work i.e. work finished in (n + 8) days Therefore, 150/150𝑛 + 146/150𝑛 + 144/150𝑛 + … + to (n + 8)term = 1 1/150𝑛 [150 + 146 + 142 + … to (n + 8)term]= 1 150 + 146 + 142 + … + to (n + 8)terms = 150n 150 + 146 + 142 + … + to (n + 8) terms this is an AP, where first term (a) = 150 Common difference = 146 – 150 = -4 We know that sum of n terms of AP Sn = 𝑛/2[2a + (n – 1)d] Putting n = n + 8 , a = 150 , d = -4 Sn + 8 = (𝑛+8)/2[2(150) + (n + 8 – 1)(-4)] = (𝑛+8)/2 [300 + (n + 7)(-4)] = (𝑛+8)/2 [300 – 4 (n + 7)] = (2(𝑛+8))/2[150 – 2(n + 7)] = (n + 8)[150 – 2(n + 7)] = (n + 8)[150 – 2n – 14] = n(150 – 2n – 14) + 1200 – 16n – 112 = 150n – 2n2 – 14n -16n + 1200 – 112 = – 2n2 + 150n – 14n – 16n + 1200 – 112 = – 2n2 + 120n + 1088 Hence, 150 +146 +142 + … to (n + 8)term = -2n2 + 120n + 1088 Also, we know that 150 +146 +142 + … to (n + 8)term = 150n – 2n2 + 120n + 1088 = 150n – 2n2 + 120n – 150n + 1088 = 0 – 2n2 – 30n + 1088 = 0 2n2 + 30n – 1088 = 0 n2 + 15n – 544 = 0 n2 + 32n – 17n – 544 = 0 n (n + 32) – 17(n + 32) = 0 (n – 17)(n + 32) = 0 Since n cannot be negative , n = – 32 is not possible Hence n = 17 Work was completed in n + 8 days i.e. 17 + 8 = 25 days

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
  • Shivam Goyal's image

    How calculate total work whose done by 150 worker? 

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  • Ayush Srivastava's image
    Ayush Srivastava
    Sept. 2, 2017, 8:25 p.m.
    21students are to be divided into three groups forming an AP three groups are to be named ENERGY SAVING','PROTECTION OF NATURAL RESOURCE GROUP', and 'HEALTH AWARENESS GROUP '. if the sum of the square of number of students in each group is 155. Find the number of students in each group ? In which group you would like to be ?
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