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Math Help - Geometric Progression

  1. #1
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    Geometric Progression

    A man borrows a sum of money from a building society and agrees to pay the loan (plus interest) over a period of years. If $A is the sum borrowed and r% the yearly rate of interest charged it can be proved that the amount (\$P_n) of each annual installment which will extinguish the loan in n years is given by the formula
    P_n=\frac{A(R-1)R^n}{R^n-1} where R=1+\frac{r}{100}
    Find in its simplest form the ratio P_{2n}:P_n. Show that this ratio is always greater than \frac{1}{2} and, if r=6.5, find the least integral number of years for which the ratio is greater than \frac{3}{5}.

    In the ratio
    P_{2n}:P_n
    \frac{A(R-1)R^{2n}}{R^{2n}-1}:\frac{A(R-1)R^n}{R^n-1}
    \frac{R^{2n}}{R^{2n}-1}:\frac{R^n}{R^n-1}
    Is this the ratio in its simplest form?
    And how do I show that it is always more than \frac{1}{2}?
    Thanks
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  2. #2
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    Quote Originally Posted by arze View Post
    A man borrows a sum of money from a building society and agrees to pay the loan (plus interest) over a period of years. If $A is the sum borrowed and r% the yearly rate of interest charged it can be proved that the amount (\$P_n) of each annual installment which will extinguish the loan in n years is given by the formula
    P_n=\frac{A(R-1)R^n}{R^n-1} where R=1+\frac{r}{100}
    Find in its simplest form the ratio P_{2n}:P_n. Show that this ratio is always greater than \frac{1}{2} and, if r=6.5, find the least integral number of years for which the ratio is greater than \frac{3}{5}.

    In the ratio
    P_{2n}:P_n
    \frac{A(R-1)R^{2n}}{R^{2n}-1}:\frac{A(R-1)R^n}{R^n-1}
    \frac{R^{2n}}{R^{2n}-1}:\frac{R^n}{R^n-1}
    Is this the ratio in its simplest form?
    And how do I show that it is always more than \frac{1}{2}?
    Thanks
    A ratio is nothing but a quotient. So you are able to simplify the last ratio by using some rules for calculating fractions:

    \frac{R^{2n}}{R^{2n}-1}:\frac{R^n}{R^n-1} = \dfrac{R^{2n}}{(R^n-1)(R^n+1)} \cdot \dfrac{R^n-1}{R^n}

    Now cancel to get \boxed{\dfrac{R^n}{R^n+1}}
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