# Math Help - Sinking fund

1. ## Sinking fund

Hi,
I’m trying to solve this problem;

A bank has three types of account in which the interest rate depends on the amount invested. The ordinary account offers a return of 6% and is available to any customer. Extra account offers 7%, only available to customers with 5,000$or more to invest, and super extra account offers 8% and is available only to customers with 20,000$ or more to invest. In each case interest is compounded annually and is added to the investment at the end of the year. A person saves 4,000$at the beginning of the year for 25 years. Calculate the total amount saved assuming the money is transferred to a higher-interest account at the earliest opportunity. Any help would be much appreciated. Thanks 2. Here is an Excel table that I made: fund.xls. Code: Year Balance Interest 1$4,000.00	0.06
2	$8,240.00 0.07 3$12,816.80	0.07
4	$17,713.98 0.07 5$22,953.95	0.08
6	$28,790.27 0.08 7$35,093.49	0.08
8	$41,900.97 0.08 9$49,253.05	0.08
10	$57,193.29 0.08 11$65,768.76	0.08
12	$75,030.26 0.08 13$85,032.68	0.08
14	$95,835.29 0.08 15$107,502.12	0.08
16	$120,102.28 0.08 17$133,710.47	0.08
18	$148,407.31 0.08 19$164,279.89	0.08
20	$181,422.28 0.08 21$199,936.06	0.08
22	$219,930.95 0.08 23$241,525.42	0.08
24	$264,847.46 0.08 25$290,035.25	0.08
26	\$313,238.07	0.08
The second column contains the balance $b_n$ at the beginning of year $n$. The third column contains the interest rate $r_n$ used at the end of year $n$. Then $b_1=4,000$ and
$b_{n+1}=
\begin{cases}
b_n(1+r_n)+4,000, & n< 25\\
b_{25}(1+r_{25}), & n=25
\end{cases}
$
and $r_{n}=
\begin{cases}
8\%, & b_n\ge 20,000\\
7\%, & 5,000\le b_n<20,000\\
6\%, & \mbox{otherwise}
\end{cases}
$

3. Thanks for the reply. The answer you got is the same as in the textbook. However this question is in preparation for an exam and I will not be able to use EXCEL. If anyone is able to produce the an answer without using excel I would be very grateful.
Cheers

4. Well, you probably need a calculator unless you are amazing at calculations, rounding and estimates. In fact, you can do all 25 iterations in reasonable time. It is also possible to calculate $b_5$, after which the interest rate stabilizes. Let $r=1+r_5=\dots=1+r_{25}=1.08$. Then

$b_6=r b_5+4000$
$b_7=r(r b_5+4000)+4000=r^2b_5+4000(r+1)$
$b_8=r(r(r b_5+4000)+4000)+4000=r^3b_5+4000(r^2+r+1)$

Following the pattern and using the formula for the sum of geometric progression,

$b_{5+k}=r^kb_5+4000(r^{k-1}+\dots+r+1)=r^k+4000(r^k-1)/(r-1)$

Thus,

$b_{25}=b_{5+20}=r^{20}b_5+4000(r^{20}-1)/(r-1)$

and $b_{26}=rb_{25}$.