Suppose an amount £200,000 is borrowed. The bank will charge interest on this loan at
\[r_1\]
%, so that after 25 years the amount of money that needs to be repaid is \[200000 (1+r_1/100)^{25}\]
If £
\[A\]
is to be invested at the start of each year, then in the first year £\[A\]
will be invested at an interest rate of \[r_2\]
.After 25 years this will have grown to
\[A(1+r_2/100)^{25}\]
In the second year £
\[A\]
will be invested at an interest rate of \[r_2\]
.After 24 years this will have grown to £
\[A(1+r_2/100)^{24}\]
in the third year £
\[A\]
will be invested at an interest rate of \[r_2\]
.After 23 years this will have grown to £
\[A(1+r_2/100)^{23}\]
Carrying on like this, in the 25th year £
\[A\]
will be invested at an interest rate of \[r_2\]
.After 1 year this will have grown to £
\[A(1+r_2/100)\]
At the end of the 25th year the last instalment of £
\[A\]
will be paid. Adding these up - in reverse order - gives\[A+A(1+r_2/100)+A(1+r_2/100)^2+...+A(1+r_2)^{25}\]
This is a geometric series with first term
\[A\]
, common ratio \[(1+r_2/100)\]
and 26 terms.The sum of the series is
\[S_{26}=\frac{A(1-(1+r_2/100)^{26}}{1-(1+r_2/100)}= \frac{100A(1-r_2)^{26}}{r_2}\]
This must be at least equal to the value of the loan after 25 years.
\[ \frac{100A(1-r_2)^{26}}{r_2} \geqslant 200000(1+r_1/100)^{25} \rightarrow A \geqslant \frac{2000r_2(1+r_1)^{25}}{(1-r_2/100)^{26}}\]