Amount Owed

Interest is charged at p.a. , compounded monthly.

So the monthly interest rate is:

The number of terms is:

So building up our terms:

Let be

Note: Generally, we build up to terms and then we are able to generalise for terms by looking at the pattern.

From this, we can formulate a GP with first term , and .

Interest Charged

Total Instalments over the six years

Reduction in loan over six years

Hence, interest charged is

Monthly Instalment

The monthly interest rate is

Building up our terms

Let be each instalment.

We can now generalise this to terms as seen below:

Formulating a GP

Thus, the loan is paid off in years so and solving for

Let be the amount owing after months.

Building up the GP

We can now generalize this to repayments:

We can now use the geometric sum formula to simplify this:

As required

Part 1 - Finding

The interest rate here is compounded monthly so:

We can now find the amount owed by working through the instalments one month at a time:

We can now generalise this to n by following the pattern. This can be seen below:

This can be simplified by applying the geometric sum formula to the series above:

as required.

Part 2 - Finding the number of months

The value of the initial loan after 15 years can be found by substituting into the equation from part 1:

Izzy’s loan is now worth

Let be the amount owing after months from the start of the new loan.

Using the development from part (1):

We can now find when the amount owing will be zero.

Amount Owed After Two Repayments

The monthly interest rate is .

Building up each term repayment:

Note: Interest is charged before each repayment

As required.

General Expression For Amount Owed

Following the trend of part (1) we get:

From this we can form a GP, where , & :

Monthly Repayment

The loan is to be repaid in years so in months:

Solving for by using part (2)

We can now let and solve for