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Math Help - Nth Partial Sum

  1. #1
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    Question Nth Partial Sum

    I start a savings account with $200 and the account earns 6% interest per year compounded annually. Every year I add an additional $200 to the account. What is the infinite series that calculates the total value of the account after n years? (at n=0 the value is $200) What is the equation for the nth partial sum (Sn)?
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by BioGrl View Post
    I start a savings account with $200 and the account earns 6% interest per year compounded annually. Every year I add an additional $200 to the account. What is the infinite series that calculates the total value of the account after n years? (at n=0 the value is $200) What is the equation for the nth partial sum (Sn)?
    The initial 200 is multiplied by 1.06^n. The second 200 is multiplied by 1.06^{n-1} and so on

    s_0=200
    s_1=200+1.06\cdot200
    s_2=200+1.06\cdot200+1.06^2\cdot200
    ...
    s_n=200\sum_{k=0}^n 1.06^k

    It's a geometric series.
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  3. #3
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    Thumbs up

    Thanx! But I have a question. Why wouldn't Sn be raised to k-1 instead of k? I am confused on how you got to the final geometric series.
    Also, if I wanted to find the value of the account after 10 years how would I do that?
    Thanx sooo much!!
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  4. #4
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by BioGrl View Post
    Thanx! But I have a question. Why wouldn't Sn be raised to k-1 instead of k? I am confused on how you got to the final geometric series.
    Also, if I wanted to find the value of the account after 10 years how would I do that?
    Thanx sooo much!!
    It should be k. If you look at the pattern going up, you can see that the highest exponent in the sum s_k of 1.06 is k.

    You can prove this by induction if you want.

    If you want to find the value after 10 years, put in n=10.
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  5. #5
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    Thanx!!!!
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