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Thread: Sum to infinity.. (don't know enough to phrase problem correctly)

  1. #1
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    Sum to infinity.. (don't know enough to phrase problem correctly)

    This problem is a basic one from finance/economics about obtaining the Present Value; but the math is general
    to any field.

    How is the value 32,097 obtained?
    Is it to do with sum to infinity?

    $\displaystyle
    PV=20,000.\sum_{i=0}^{infinity}\frac{1}{(1+0.05)^{ 20i}}=32,097
    $


    I haven't given any context (just a word problem). If needed I will provide.

    I need help with this; any help is really appreciated!
    thanks
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  2. #2
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    Quote Originally Posted by needlittlehelp01 View Post
    This problem is a basic one from finance/economics about obtaining the Present Value; but the math is general
    to any field.

    How is the value 32,097 obtained?
    Is it to do with sum to infinity?

    $\displaystyle
    PV=20,000.\sum_{i=0}^{infinity}\frac{1}{(1+0.05)^{ 20i}}=32,097
    $


    I haven't given any context (just a word problem). If needed I will provide.

    I need help with this; any help is really appreciated!
    thanks

    You have the sum of an infinte geometric series: if $\displaystyle a,ar,ar^2,\ldots,ar^n\ldots$ is an infinite sequence with first element $\displaystyle a$ and common ratio $\displaystyle r$ , and such that $\displaystyle |r|<1$ , then its infinite sum is:

    $\displaystyle \sum^\infty_{k=0}ar^k=\frac{a}{1-r}$

    Now you first recognize the geometric series, recognize $\displaystyle a\,\,\,and\,\,\,r$ , and then do the infinite sum. You'll check at once that the result indeed is $\displaystyle 32,097.0348...$ , which they rounded up to $\displaystyle 32,097$.

    This stuff is usually learned in high school, and it definitely doesn't belong to abstract algebra.

    Tonio
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  3. #3
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    Quote Originally Posted by needlittlehelp01 View Post
    This problem is a basic one from finance/economics about obtaining the Present Value; but the math is general
    to any field.

    How is the value 32,097 obtained?
    Is it to do with sum to infinity?

    $\displaystyle
    PV=20,000.\sum_{i=0}^{infinity}\frac{1}{(1+0.05)^{ 20i}}=32,097
    $




    I haven't given any context (just a word problem). If needed I will provide.

    I need help with this; any help is really appreciated!
    thanks
    First, this is in the wrong section- it has nothing to do with "Linear and Abstract Algebra".

    Second, that is a "geometric series"- it is of the form $\displaystyle a\sum r^n$ with a= 20000 and $\displaystyle r= \frac{1}{1.05^{20}}= 0.376889$, approximately.

    A geometric series has sum $\displaystyle \frac{a}{1- r}$ which, here, is $\displaystyle \frac{20000}{1- 0.376889}= 32097$

    Once again, Tonio gets in just ahead of me!
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    ...........

    Once again, Tonio gets in just ahead of me!


    Tonio
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