# 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 $A is the sum borrowed and r% the yearly rate of interest charged it can be proved that the amount$\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
• 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 $A is the sum borrowed and r% the yearly rate of interest charged it can be proved that the amount$\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

A ratio is nothing but a quotient. So you are able to simplify the last ratio by using some rules for calculating fractions:

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

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