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Math Help - sum of arithmetic sequence

  1. #1
    Senior Member furor celtica's Avatar
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    sum of arithmetic sequence

    A woman started a business with a workforce of 50 people. Every two weeks the number of people in the workforce is increased by 3 people.
    How many people were there in the workforce after 26 weeks (I answered 89, which is correct.)
    Each member of the workforce earned $600 per week. What was the total wage bill for this 26 weeks?

    I used the formula for the sum of an arithmetic sequence S=0.5n(a+l) and (for convention) 'eliminated' every other week where the wage bill remained the same to come up with S(half the wage bill) = 0.5 x 13 x (30 000 + 51 600)
    So the total wage bill is 13 x 81 600 = $ 1 060 800

    However the textbook gives the answer to be $ 1 341 600
    Where did I go wrong?
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  2. #2
    Member kalyanram's Avatar
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    Re: sum of arithmetic sequence

    Hello Furor,
    I agree with your result for the wages. I think the result given in the text book is incorrect. Here is how it is
    I would like to suggest the following observation
    wages Weeks
    30000 1-2
    30000 +1800 2-4
    30000 +2*1800 4-6
    :
    :
    :
    sum of wages = 26*(30000) + 3600*(0+1+....+12) = 780000+ 3600*78 = 1060800

    Now one thing that we can fairly agree is the fact that the 26*30000 component is the component that has to be paid as 50 workers are present from the beginning

    Lets deduct that from both the numbers

    1060800 - 780000 = 280800
    1341600 - 780000 = 561600

    So we see that 561600 = 2*280800 which essentially tells us that a miscalculation has been done in terms of halving the number of weeks.

    Try to verify my calculations in case I made a mistake some where.

    Kalyan
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  3. #3
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    Re: sum of arithmetic sequence

    Quote Originally Posted by furor celtica View Post
    A woman started a business with a workforce of 50 people. Every two weeks the number of people in the workforce is increased by 3 people.
    How many people were there in the workforce after 26 weeks (I answered 89, which is correct.)
    Each member of the workforce earned $600 per week. What was the total wage bill for this 26 weeks?

    I used the formula for the sum of an arithmetic sequence S=0.5n(a+l) and (for convention) 'eliminated' every other week where the wage bill remained the same to come up with S(half the wage bill) = 0.5 x 13 x (30 000 + 51 600)
    So the total wage bill is 13 x 81 600 = $ 1 060 800

    However the textbook gives the answer to be $ 1 341 600
    Where did I go wrong?
    Hello!
    For the first part of the question, I assume you did 50 + (3 people x 13 weeks) = 89 people, because every 2 weeks there is an increase of 3 people. Which is correct.

    Next part, because I have no idea the arithmetic sequence, i'll use the long method.

    50 people x 600$ x 26 weeks = 780,000 I did this because 50 people is a constant.

    3 people x 600$ x 24 weeks = 43200
    3 people x 600$ x 22 weeks = 39600
    3 people x 600$ x 20 weeks = 36000, Now the pattern decreases by 3600 so..
    Week 18 = 32400
    Week 16 = 28800
    Week 14 = 25200
    Week 12 = 21600
    Week 10 = 18000
    Week 8 = 14400
    Week 6 = 10800
    Week 4 = 7200
    Week 2 = 3600

    Add the numbers to get .. 280800
    Add the 2 numbers to arrive at 280,800 + 780,000 = $1 060 800.

    Either the book is wrong, or you and I both did something wrong. Good luck.
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