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anonymous

  • one year ago

Tom has taken out a loan for college. He started paying off the loan with a first payment of $200. Each month he pays, he wants to pay back 1.2 times as the amount he paid the month before. Explain to Tom how to represent his first 30 payments in sigma notation. Then explain how to find the sum of his first 30 payments, using complete sentences. Explain why this series is convergent or divergent

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  1. jim_thompson5910
    • one year ago
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    hint: this is a geometric sequence with a = 200 being the first term r = 1.2 being the common ratio

  2. anonymous
    • one year ago
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    What was the formula for a geometric sequence again? I'm doing a lot of essay questions that need to be done in 20 minutes lol

  3. jim_thompson5910
    • one year ago
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    The nth term of a geometric sequence is \[\Large a(r)^{n-1}\]

  4. anonymous
    • one year ago
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    And n would be the number of payments correct?

  5. jim_thompson5910
    • one year ago
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    n is the nth payment so say n = 3, that would represent the 3rd payment. To find the actual 3rd payment, plug n = 3 into the nth term expression I wrote above

  6. jim_thompson5910
    • one year ago
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    a = 200 r = 1.2 those two values above are going to stay fixed like that

  7. anonymous
    • one year ago
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    Okay so it would be \[200(1.2)^{3-1}=344.6\]

  8. jim_thompson5910
    • one year ago
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    you should get this \[\Large 200(1.2)^{3-1}=288\] so that means payment n = 3 is $288

  9. jim_thompson5910
    • one year ago
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    the other values of n are calculated in a similar way

  10. anonymous
    • one year ago
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    My calculator sucks haha. I got it now. If we substitute 4 for 3 we'd get \[200(1.2)^{4-1}=414.72?\] or am I doing it wrong?

  11. jim_thompson5910
    • one year ago
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    it's incorrect

  12. jim_thompson5910
    • one year ago
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    the "4-1" turns into "3"

  13. jim_thompson5910
    • one year ago
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    if you are typing into a calculator, make sure to use parenthesis 200*(1.2)^(4-1)

  14. jim_thompson5910
    • one year ago
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    http://web2.0calc.com/#200*(1.2)^(4-1)

  15. anonymous
    • one year ago
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    Would it be 345.6?

  16. jim_thompson5910
    • one year ago
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    yes

  17. jim_thompson5910
    • one year ago
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    so imagine this happening all the way from n = 1 to n = 30 we could write out every single term, but that's very tedious and not necessary with proper math notation we can use sigma notation to represent a shortcut which essentially says "add up a bunch of these terms" |dw:1433563109879:dw| that funky looking E is the capital greek letter sigma

  18. jim_thompson5910
    • one year ago
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    it means "add up a bunch of terms" |dw:1433563144808:dw|

  19. jim_thompson5910
    • one year ago
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    |dw:1433563185734:dw|

  20. jim_thompson5910
    • one year ago
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    |dw:1433563205308:dw|

  21. jim_thompson5910
    • one year ago
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    The expression \[\Large 200(1.2)^{k-1}\] will go to the left of the sigma |dw:1433563248671:dw|

  22. jim_thompson5910
    • one year ago
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    so the whole drawing of this |dw:1433563286174:dw| means "add up 30 copies of 200*(1.2)^(k-1)" where k will vary from k = 1 all the way up to k = 30

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