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Endowment Mortgages

Anyone taking out an endowment mortgage to buy a house borrows money from a bank promising to repay the loan after a 25 or so years, and raises the money to repay the loan by investing a certain amount of money each year, the amount being calculated so that after 25 years, the total investment will pay off the loan.
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.
\[ \frac{100A(1-r_2)^{26}}{r_2} \geqslant 200000(1+r_1/100)^{25} \rightarrow A \geqslant \frac{2000r_2(1+r_1)^{25}}{(1-r_2/100)^{26}}\]