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Sequence/series math problem

  1. Nov 20, 2006 #1
    n = number of items

    pay scale #1
    tn = n$0.65
    so if n = 10, tn = $6.50

    pay scale #2
    $0.10 for each item, plus 0.05 for each additional item. so for the 2nd sale she'd make 0.15, third she'd make 0.20, etc.

    if tn = 1. then it = 0.10
    so 2 = 0.15 + 10 = 0.25
    3 = 0.20 + 0.25 = 0.40
    4 = 0.30 + 0.40 = 0.70

    and so on..

    so whats the formula for pay scale #2?
    for the first part of the formula i have:
    tn = 0.10 + 0.05n-1 and then to that i have to add the previous number (n-1)?

    and at what point is pay scale #2 better than pay scale #1 (i got after 23 using the amy-method)

    any help is appreciated!

  2. jcsd
  3. Nov 20, 2006 #2
    but i have to add the total amount together. not just price per item. total amount.

  4. Nov 20, 2006 #3
    [tex] T_{n} = 0.10n +0.05(n-1) [/tex] starting from n = 1.

    [tex] S_{n} = \frac{n}{2}(a_{1}+a_{n})[/tex]

    [tex] S_{n} = \left \frac{n}{2}(0.10+ 0.10n +0.05(n-1))\right[/tex]

    Set [tex] \frac{n}{2}(0.10+ 0.10n +0.05(n-1) = 0.65n [/tex] and solve for [tex] n [/tex]
    Last edited: Nov 20, 2006
  5. Nov 21, 2006 #4
    thanks for the help, but it's still a little unclear to me.
    what do you mean by 'set'? and what variable should i use for n? is 23 the right answer for when pay scale 2 is better than payscale 1?

    btw: how hard would you say this question is? (grade level). i asked a high school math teacher yesterday and he didnt have the foggiest idea.

  6. Nov 21, 2006 #5


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    Hi Amy,

    By set courtrigard means 'make' or 'let'. In other words

    "Let the LHS be equal to the RHS"

    As courtirgard says above, you shouldn't be substituting anything for n, you should be solving for n.
  7. Nov 21, 2006 #6
    By "set" I meant solve the following equation for [tex] n [/tex]: [tex] \frac{n}{2}(0.10+ 0.10n +0.05(n-1) = 0.65n [/tex]
  8. Nov 21, 2006 #7


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    Apologies for butting in courtigrad :redface:
  9. Nov 21, 2006 #8
    Wait, so...
    Pay Scale 1:
    First item costs .65, second item costs .65, third item costs .65, etc.
    Pay Scale 2:
    First item costs .10, second item costs .15, third item costs .20, etc.

    Is this what the question means?
  10. Nov 21, 2006 #9


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    No, not quite. Look at the question again, taking note of where the term n appears.
  11. Nov 21, 2006 #10
    yes. but you have to figure out what pay scale is the better deal. like i calculated after 23 items pay scale #1 pays $14.95, payscale #2 pays $15. so for 23 items and over payscale #2 is the better deal. (according to the amy-method)

    should i write the question exactly how it appears in the book? (maybe im not making sense of it either) :blushing:

    i see what you mean be solving for n, but dunno how.

  12. Nov 21, 2006 #11
    I get 13.80 for 23 items on Pay Scale 2.
  13. Nov 21, 2006 #12


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    There's a famous story about the mathematician Gauss who was given the assignment to add all of the numbers between 1 and 100 by a grade school teacher that was annoyed with Gauss. The teacher was rather incredulous when Gaus provided the answer in less than a minute.

    Now, Gauss is considered one of the greatest mathematicians of all time, so expecting students in elementary school to be able to figure this out on their own is probably a bit of a push, but I was in the third grade when it was explained to me, and it made plenty of sense to me then.

    For payscale #2, the mean value per item is always going to be the same as the value of the middle item (for an odd number of items) or as the mean value of the middle two items (for an even number of items). That is because the items outside the middle can be symetrically paired up with each pair having the same average. Now, it's easy to figure out the number items, and the pay per item. The closed form for your example is:

    [tex]p(n)=\left(\frac{(n-1)}{2}\times 5 + 10 \right) \times n[/tex]

    Where the expression inside the parentheses is the average pay per item.
    Last edited: Nov 21, 2006
  14. Nov 21, 2006 #13
    that was my answer for 22 items.

    Nate. i tried your formula for the few numbers. like for 10 units, i ended up with 325. somethings missing.

    here's the exact question:

    "mabel is offered a job selling magazine substriptions. she has the choice of two pay scales.

    Pay scale 1:
    she can be paid $0.65 for each subscription she sells.

    Pay scale 2:
    she can be paid $0.10 for the first subscription, with the wage going to $0.05 more for each subscription after the first. for her first sale she would make $0.10, for her second sale she would make $0.15, for her third sale she would make $0.20, and so on.

    analyze and compare the two pay scales. which pay scale is better? explain your reasoning. include the general term and some specific calculations in your analysis".

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