Note: Defining all the information in terms of variables should always be done before starting the questions as it makes your working out cleaner and less prone to mistakes.

Let Amount in superannuation after compounding periods (years)

Let Instalment amount

Let Number of compounding periods

Let Interest per compounding period

Solution to (1)

Step 1 - Setup variables

Step 2 - Build the annuity for after one year (one compounding period),

Step 3 - Substitute values

Solution to (2)

Step 1 - Setup variables

Step 2 - Build the annuity for after two years (Use the expression for from the previous part)

Step 3 - Substitute values

Solution to (3)

Step 1 - Setup variables

Step 2 - Build the annuity for after three years (Use the expression for from the previous part)

Step 3 - Substitute values

Let Amount in superannuation after compounding periods (years)

Let Instalment amount

Let Number of compounding periods

Let Interest per compounding period

Solution for (1)

Step 1 - Setup variables

Step 2 - Build the annuity for after years

Therefore,

From here we can apply the sum of a geometric series to the series of terms enclosed within the square brackets,

Number of terms within the square brackets terms

Step 3 - Substitute values

Solution for (2)

Step 1 - Setup variables

Step 2 - Substitute into the expression and solve,

Let Total amount of investment after compounding periods (years)

Let Instalment amount

Let Number of compounding periods

Let Interest per compounding period

Let P = Initial contribution amount

Step 1 - Setup variables

Step 2 - Build the annuity for years

Sum the geometric series within the square brackets,

Step 3 - Substitute values

Step 1 - Setup variables

Let

Let $200

Let (we divide the rate per annum by in order to obtain the rate per month, as the compounding period is monthly)

Step 2 - Build an expression for the investment

Note: There is no interest accumulated at the end of the first month, because William invests $200 at the end of the month, not at the start.

Step 3 - Make the subject

By index laws, we can bring the out of the as shown,

Step 4 - Substitute values

Note: We need to round up, as rounding down to will result in a final investment value of less than .

Therefore the number of years for the investment value to reach is years .

Solution to (1)

Step 1 - Setup variables

Let Amount in account after compounding periods (weeks)

Let

Let

Step 2 - Build an expression for the investment after weeks

Step 3 - Substitute values

Solution to (2)

Step 1 - Setup variables

Let Amount in account after compounding periods (weeks)

Let Amount in account after compounding periods

Let

Let , which is the contribution amount after the week.

Let

Step 2 - Determine amount in account after weeks

Step 3 - Form an expression for the next weeks

Step 4 - Substitute values

Therefore, after weeks, the account has in total.