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Math Help - Compound interest

  1. #1
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    Compound interest

    A deposit of $25 is made at the beginning of the 1st month, and successive monthly deposits after that is $25 more than the previous month (2nd month is a $50 deposit, 3rd month is a $75, etc.). At the beginning of the next year (after 12 months), the deposit cycle is reset back to $25 the first month, etc. and this pattern continues for 5 years. The account pays 5% compounded interest monthly at the end of each month. What is the balance of the account after 5 years?
    So I'm trying to find out the formula by writing out the expanded version first and simplifying it, but I'm not sure how to write this in terms of exponents.

    Month 1: 25 + 25(.05)

    Let x = 25 + 25(.05)

    Month 2: (x + 50)(.05) + (x + 50)

    Month 3: ((x + 50)(.05) + (x + 50) + 75)(.05) + ((x + 50)(.05) + (x + 50) + 75)

    Month 4: (((x + 50)(.05) + (x + 50) + 100)(.05) + ((x + 50)(.05) +  (x + 50) + 100))(.05) + ((x + 50)(.05) + (x + 50) + 100)(.05) + ((x +  50)(.05) + (x + 50) + 100)

    But then I remembered the formula will probably change after 12 months since the deposits start over, but the balance is different...so I'm not sure how else to really approach this.
    Last edited by daigo; July 4th 2012 at 07:42 AM.
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  2. #2
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    Re: Compound interest

    So far, this is what I got for the first year:

    (25m)(1 + \frac{.05}{12})(\frac{1 - (1 + \frac{.05}{12})^{12}}{{1 - (1 + \frac{.05}{12})}})
    where m = the month...

    Not sure if this is correct, but do I need to make a new formula for each year to make up for the "new" principal in the account (the total from the preceding year[s])?
    Last edited by daigo; July 4th 2012 at 10:38 AM.
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  3. #3
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    Re: Compound interest

    Daigo, I'll give you the account balance after 5 years: 11,012.44
    Deposited is $1,950 per year, total $9,750 : 11012.44 - 9750 = 1262.44 is interest.

    The stuff you've shown makes no sense (sorry!), so I'm not going to try and show you
    how to get this by formula...why? Because I don't know where you're at with annuities.

    Here's a deal:
    $25 is deposited monthly for 12 months.
    The 1st deposit is at end of 1st month.
    The rate is 5% annual compounded monthly.
    What is the accont balance at end of 12th month?

    Show me the solution, and HOW you got it.
    Then I'll have a better idea of how to help. How's that?!
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  4. #4
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    Re: Compound interest

    For your problem:

    Month 1: $0 principal + $25 deposited
    5% compounded interest of $25 = $26.25

    Month 2: $26.25 principal + $25 deposited
    5% compounded interest of $51.25 = $53.8125

    Month 3: $53.8125 principal + $25 deposited
    5% compounded interest of $78.8125 = $82.753125

    Month 4: $82.753125 principal + $25 deposited
    5% compounded interest of $107.753125 = $113.140781

    Month 5: $113.140781 principal + $25 deposited
    5% compounded interest of $138.140781 = $145.04782

    Month 6: $145.04782 principal + $25 deposited
    5% compounded interest of $170.04782 = $178.550211

    Month 7: $178.550211 principal + $25 deposited
    5% compounded interest of $203.550211 = $213.727722

    Month 8: $213.727722 principal + $25 deposited
    5% compounded interest of $238.727722 = $250.664108

    Month 9: $250.664108 principal + $25 deposited
    5% compounded interest of $275.664108 = $289.447313

    Month 10: $289.447313 principal + $25 deposited
    5% compounded interest of $314.447313 = $330.169679

    Month 11: $330.169679 principal + $25 deposited
    5% compounded interest of $355.169679 = $372.928163

    Month 12: $372.928163 principal + $25 deposited
    5% compounded interest of $397.928163 = $417.824571

    So after 1 year the total is $417.824571 with 5% compounded monthly interest if you make a $25 deposit each month





    I might as well do the 1st year for my question too:

    Beginning of month 1: $0 principal + $25 deposited
    End of month 1: 5% compounded interest of $25 = $26.25

    Beginning of month 2: $26.25 principal + $50 deposited
    End of month 2: 5% compounded interest of $76.25 = $80.0625

    Beginning of month 3: $80.0625 principal + $75 deposited
    End of month 3: 5% compounded interest of $155.0625 = $162.815625

    Beginning of month 4: $162.815625 principal + $100 deposited
    End of month 4: 5% compounded interest of $262.815625 = $275.956406

    Beginning of month 5: $275.956406 principal + $125 deposited
    End of month 5: 5% compounded interest of $400.956406 = $421.004226

    Beginning of month 6: $421.004226 principal + $150 deposited
    End of month 6: 5% compounded interest of $571.004226 = $599.554437

    Beginning of month 7: $599.554437 principal + $175 deposited
    End of month 7: 5% compounded interest of $774.554437 = $813.282159

    Beginning of month 8: $813.282159 principal + $200 deposited
    End of month 8: 5% compounded interest of $1,013.28216 = $1,063.94627

    Beginning of month 9: $1,063.94627 principal + $225 deposited
    End of month 9: 5% compounded interest of $1,288.94627 = $1,353.39358

    Beginning of month 10: $1,353.39358 principal + $250 deposited
    End of month 10: 5% compounded interest of $1,603.39358 = $1,683.56326

    Beginning of month 11: $1,683.56326 principal + $275 deposited
    End of month 11: 5% compounded interest of $1,958.56326 = $2,056.49142

    Beginning of month 12: $2,056.49142 principal + $300 deposited
    End of month 12: 5% compounded interest of $2,356.49142 = $2,474.31599

    And then at the beginning of year 2 the deposits reset:

    Beginning of month 1: $2,474.31599 principal + $25 deposited
    End of month 1: 5% compounded interest of $2,499.31599 = $2,624.28179

    etc.
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  5. #5
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    Re: Compound interest

    Quote Originally Posted by daigo View Post
    For your problem:
    Month 1: $0 principal + $25 deposited
    5% compounded interest of $25 = $26.25
    ..........
    Month 12: $372.928163 principal + $25 deposited
    5% compounded interest of $397.928163 = $417.824571

    So after 1 year the total is $417.824571 with 5% compounded monthly interest if you make a $25 deposit each month
    You're somewhat on right track...but:
    Month 1 is $25 (no interest).
    You have to calculate interest using .05/12, not .05 : .05 each month means a 60% annual rate!
    Works out like this:
    Code:
    M   DEPOSIT  INTEREST   BALANCE
    0                           .00
    1     25.00       .00     25.00
    2     25.00       .10     50.10
    3     25.00       .21     75.31
    4     25.00       .31    100.62
    ...
    12    25.00      1.17    306.97
    Formula:
    d = 25
    n = 12
    i = .05/12
    F = ?
    F = d[(1+i)^n - 1] / i = 306.97
    Last edited by Wilmer; July 4th 2012 at 12:21 PM.
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  6. #6
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    Re: Compound interest

    Wouldn't a 60% annual rate be the interest for the entire year?

    i.e. ($25 deposit per month x 12 months) + ($25 deposit per month x 12 months)(.6 interest)

    But anyway, trying to do the way you showed it:

    Month 2:
    $25 principal + ($25 principal * (.5/12 interest)) + $25 deposit = $51.0416667

    Month 3:
    $51.0416667 + ($51.0416667 * (.5/12)) + $25 deposit = $78.1684028

    I'm still not getting the numbers you are getting...what am I doing wrong?
    Last edited by daigo; July 4th 2012 at 12:40 PM.
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  7. #7
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    Re: Compound interest

    Quote Originally Posted by daigo View Post
    Month 2:
    $25 principal + ($25 principal * (.5/12 interest)) + $25 deposit = $51.0416667
    Please be careful! .05/12, not .5/12
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  8. #8
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    Re: Compound interest

    Never mind, I think I finally get what "compounding" actually means now. I had been thinking differently before.

    But how do I make each deposit increment by +25 from the previous month and add it to the current month?
    Last edited by daigo; July 4th 2012 at 01:48 PM.
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