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Part 1 – Numerical Sequences

A recursive equation is said to be of first order if the value of a member depends only on the immediately preceding member. For example:

u(n + 1) = 3u(n) − 7 is first order

u(n + 1) = 3u(n) − 5u(n − 1) is not first order A first order linear recursive equation has the form

u(n + 1) = mu(n) + c, where m and c are constants.

Task 1: recursive equation

u(0) = 1,

For various values of c,

1. Choose three different values of c and plot the resultant sequences.

2. Comment on the behavior of the sequences using text boxes.

3. By assuming that the sequence does converge and letting u(n) → A as n → ∞, show how the limit can be obtained without working through the iterations.

Task 2: recursive equation.

u(0)=2,

For various of m,

1. For Task 2 (i), investigate the recurrence relation for m ≤ −1.

• You will require at least two values of m < −1,

• You will require the case where m = −1.

• Describe the behaviour of the sequences and plot them.

2. For Task 2 (ii), investigate the recurrence relation for −1 < m < 1.

• You will require at least two values of m such that −1 < m < 0,

• You will require the case where m = 0.

• You will require at least two values of m such that 0 < m < 1,

• Describe the behaviour of the sequences and plot them. 3.For Task 2 (iii), investigate the recurrence relation for m ≥ 1.

• You will require at least two values of m > 1,

• You will require the case where m = 1.

• Describe the behaviour of the sequences and plot them.

Part 2: Sequences in Finance.

Task 3

An ISA bank account pays 1.05%, compounded annually.

1. Complete this section by hand calculation.

• John makes an initial deposit of £2000 on 1st January 2016 and plans to leave the money in the ISA for the next five years, with no further deposits or withdrawals.

• Calculate the amount in John’s ISA on each 1st January for the next five years.

• Write down a recursive equation for the amount in his account after n years.

2. Complete this section using EXCEL

• Program your recursive equation using EXCEL, rounding to the nearest penny. If possible, program your equation so that changing the interest rate is done separately from the equation. Save your EXCEL file, with the sheet labelled “Task 3”.

• Use this to check your calculations in section (1) and provide commentary using text boxes.

• Illustrate your EXCEL model with a suitable graph.

3. Use your spreadsheet to answer the following questions:

• If the annual interest rate from 1st January 2016 fell to 0.5% and remained at this level over the five years, how much would be in the ISA on 1st January 2021?

• If the annual interest rate from 1st January 2016 rose to 1.3% and remained at this level over the five years, how much would be in the ISA on 1st January 2021?

4. By hand calculation, after how many years will John be able to afford a £2500 deposit on a house (at 1.05% compound interest)? Verify your answer using your program.

5. By hand calculation, work out to 3 decimal places, what interest rate is required for John to have £3000 on 1st January 2021. Verify your answer using your program.

Task 4

A second bank account pays 1.6% interest each year, but also takes away £8 per year as a service charge (after first adding on the interest).

1. Complete this section by hand calculation

• If John makes an initial deposit of £2000 and leaves the money in the ISA for the next five years, with no further deposits or withdrawals, how much will he have?

• Write down a recursive equation for the amount in his account after n years.

2. Complete this section using EXCEL

• Program your recursive equation using EXCEL, rounding to the nearest penny. If possible, program your equation so that changing the initial deposit is done separately from the equation.

• Use this to check your calculations in section (1) and provide commentary using text boxes.

• Illustrate your EXCEL model with a suitable graph and save your EXCEL sheet labelled “Task 4”.

3. By hand calculation, work out what initial deposit would mean that John neither makes, nor loses money. Verify your answer using your program.