anonymous
  • anonymous
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
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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jim_thompson5910
  • jim_thompson5910
hint: this is a geometric sequence with a = 200 being the first term r = 1.2 being the common ratio
anonymous
  • anonymous
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
jim_thompson5910
  • jim_thompson5910
The nth term of a geometric sequence is \[\Large a(r)^{n-1}\]

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anonymous
  • anonymous
And n would be the number of payments correct?
jim_thompson5910
  • jim_thompson5910
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
jim_thompson5910
  • jim_thompson5910
a = 200 r = 1.2 those two values above are going to stay fixed like that
anonymous
  • anonymous
Okay so it would be \[200(1.2)^{3-1}=344.6\]
jim_thompson5910
  • jim_thompson5910
you should get this \[\Large 200(1.2)^{3-1}=288\] so that means payment n = 3 is $288
jim_thompson5910
  • jim_thompson5910
the other values of n are calculated in a similar way
anonymous
  • anonymous
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?
jim_thompson5910
  • jim_thompson5910
it's incorrect
jim_thompson5910
  • jim_thompson5910
the "4-1" turns into "3"
jim_thompson5910
  • jim_thompson5910
if you are typing into a calculator, make sure to use parenthesis 200*(1.2)^(4-1)
jim_thompson5910
  • jim_thompson5910
http://web2.0calc.com/#200*(1.2)^(4-1)
anonymous
  • anonymous
Would it be 345.6?
jim_thompson5910
  • jim_thompson5910
yes
jim_thompson5910
  • jim_thompson5910
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
jim_thompson5910
  • jim_thompson5910
it means "add up a bunch of terms" |dw:1433563144808:dw|
jim_thompson5910
  • jim_thompson5910
|dw:1433563185734:dw|
jim_thompson5910
  • jim_thompson5910
|dw:1433563205308:dw|
jim_thompson5910
  • jim_thompson5910
The expression \[\Large 200(1.2)^{k-1}\] will go to the left of the sigma |dw:1433563248671:dw|
jim_thompson5910
  • jim_thompson5910
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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