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Math Help - Geometric Series Help

  1. #1
    Newbie
    Joined
    Dec 2008
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    2

    Exclamation Geometric Series Help

    Hello,
    I have a math problem that I cannot do, I have spent the past 3 hours on it and got relatively no where!!

    The question: A mortgage is taken out for 80000. It is to be paid by annual instalments of 5000 with the first payment being made at the end of the first year that the mortgage was taken out. Interest of 4% is then charged on any outstanding debt. Find the total time taken to pay off the mortgage.

    HINT: Find an expression for the debt remaining after n years and solve using the fact that if it is paid off, the debt = 0.

    What I started with: debt left = (80000 - 5000n)*1.04 - does not work
    What I got up to was: After nth year debt left was = 1.04*(n-1 value - 5000)
    Didn't not get anywhere with that either.
    Please help!!!
    Thanks
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  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
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    615
    Hello, person!

    This confusion has turned up from time to time.

    I don't know who is teaching that a time-payment plan is a simple geometric series.
    It is very annoying and I wish they would stop!


    A mortgage is taken out for $80,000.
    It is to be paid by annual instalments of $5000 with the first payment being made
    at the end of the first year that the mortgage was taken out.
    Interest of 4% is then charged on any outstanding debt.
    Find the total time taken to pay off the mortgage.

    HINT: Find an expression for the debt remaining after n years
    and solve using the fact that if it is paid off, the debt = 0.

    This is an Amortization problem.
    You can (1) derive the Amortization formula, or (2) memorize the formula.
    . . Neither task is pleasant.


    Let: . \begin{array}{ccc}P &=& \text{Principal borrowed} \\ r &=& \text{Periodic interest rate} \\ n &=& \text{No. of periods} \\ A &=& \text{Periodic payment} \end{array}


    We borrow P dollars at time zero.


    In one year, they charge r% interest.
    At the end of year-1, we owe: . P(1+r) dollars.
    Then we pay A dollars.
    . . Our balance is: . P(1+r) - A dollars.


    During year-2, they charge r% interest.
    At the end of year-2, we owe: . (1+r)\bigg[P(1+r) - A\bigg] dollars.
    Then we pay A dollars.
    . . Our balance is: . (1+r)^2P - (1+r)A - A dollars.


    During year-3, they charge r% interest.
    At the end of year-3, we owe: . (1+r)\bigg[(1+r)^2P - (1+r)A - A\bigg] dollars.
    Then we pay A dollars.
    . . Our balance is: . (1+r)^3P - (1+r)^2A - (1+r)A - A dollars.


    See the pattern?


    At the end of year- n, our balance is:
    . . (1+r)^nP - (1+r)^{n-1}A - (1+r)^{n-2}A - \hdots - A dollars.

    But by year- n, we expect to pay off the loan; the balance is $0.


    So, we have: . (1+r)^nP -(1+r)^{n-1}A - (1+r)^{n-2} - \hdots - A \;=\;0

    . . . . (1+r)^nP \;=\;A\underbrace{\bigg[1 + (1+r)^2 + (1+r)^3 + \hdots + (1+r)^{n-1}\bigg]}_{\text{geomtric series}} .[1]


    The geometric series has the sum: . S \;=\;\frac{(1+r)^n - 1}{(1+r)-1} \;=\;\frac{(1+r)^n-1}{r}

    Then [1] becomes: . (1+r)^nP \;=\;A\cdot\frac{(1+r)^n-1}{r}

    Therefore: . \boxed{A \;=\;P\,\frac{r(1+r)^n}{(1+r)^n-1}} . . . the Amortization Formula !


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    Back to the problem . . .


    We are given: . P = 80,\!000,\;A = 5,\!000,\;r = 4\% = 0.04

    Substitute into the Formula: . 5,\!000 \;=\;80,\!000\,\frac{(0.04)(1.04^n)}{1.04^n-1}

    . . . . 1.04^n-1 \;=\;0.64(1.04^n) \quad\Rightarrow\quad 1.04^n -0.64(1.04^n) \;=\;1

    . . . . 0.36(1.04^n) \;=\;1\quad\Rightarrow\quad 1.04^n \;=\;\tfrac{1}{0.36}


    Take logs: . \ln\left(1.04^n\right) \;=\; \ln\left(\tfrac{1}{0.36}\right) \quad\Rightarrow\quad n\ln(1.04) \;=\;\ln(\tfrac{1}{0.36})


    Therefore: . n \;=\;\frac{\ln(\frac{1}{0.36})}{\ln(1.04)} \;=\;26.04876774 \;\approx\; 26 years.


    I need a nap . . .
    .
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  3. #3
    Newbie
    Joined
    Dec 2008
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    Thank you!!!!!! That was longer than I had expected and much harder than thought I have not learnt that formula lol, take your nap and have some rep!
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