# Geometric Progression

• Dec 20th 2009, 09:08 PM
arze
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 • Dec 20th 2009, 10:16 PM earboth Quote: Originally Posted by 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 $(\P_n)$ 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 $\boxed{\dfrac{R^n}{R^n+1}}$