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**arze** 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 $\displaystyle (\$P_n)$ of each annual installment which will extinguish the loan in n years is given by the formula

$\displaystyle P_n=\frac{A(R-1)R^n}{R^n-1}$ where $\displaystyle R=1+\frac{r}{100}$

Find in its simplest form the ratio $\displaystyle P_{2n}:P_n$. Show that this ratio is always greater than $\displaystyle \frac{1}{2}$ and, if r=6.5, find the least integral number of years for which the ratio is greater than $\displaystyle \frac{3}{5}$.

In the ratio

$\displaystyle P_{2n}:P_n$

$\displaystyle \frac{A(R-1)R^{2n}}{R^{2n}-1}:\frac{A(R-1)R^n}{R^n-1}$

$\displaystyle \frac{R^{2n}}{R^{2n}-1}:\frac{R^n}{R^n-1}$

Is this the ratio in its simplest form?

And how do I show that it is always more than $\displaystyle \frac{1}{2}$?

Thanks