Recurrence Relations

Recurrence relations are a further method of modelling growth.

They are used to predict the next value in a number pattern

The relation tells you how to get from one value to the next.

Un Notation for number patterns

U1= value of 1st term

U2 = value of 2nd term etc

Un = value of nth term

Un–1 is the term before Un and Un+1 is the term after Un

In the number pattern

1, 4, 7, 10, 13, 16

U1= 1 and U4= 10

To find the next number in the pattern add 3 to the term before.

U2 = U1 + 3 = 1 + 3 = 4

U3 = U2 + 3 = 4 + 3 = 7

U4 = U3 + 3 = 7 + 3 = 10

To find the next term add 3 to the term before.

nth term = (n–1)th term + 3

Un = Un–1 + 3

Next term = term before + 3

So if n = 4 then U4 = U3 + 3

This could have been written as

Un+1 = Un + 3

Next term = term before + 3

So if n = 4 then U5 = U4 + 3

Un = 2Un–1 + 3 U1 = 3

Next term = 2term before + 3

U1 = 3

U2 = 2U1 + 3 = 23 + 3 = 9

U3 = 2U2 + 3 = 29 + 3 = 21

U4 = 2U3 + 3 = 221 + 3 = 45

In questions involving investing money at a certain interest rate U0 is usually used for the initial investment so that U1 will give the value after 1 month or 1 year

The pattern Pn+1 = 1.05Pn is in fact increasing the previous value by 5% so we can find out the value of £10 if it is invested at an annual interest rate of 5%.

Pn+1 = 1.05Pn P0 = 10

Next term = 1.05term before

P0 is used in this case to show that P1 is the value after 1 year

P0 = 10

P1 = 1.05P0 = 1.0510 = 10.50

P2 = 1.05P1 = 1.0510.5 = 11.025

P3 = 1.05P2 = 1.0511.025 = 11.57625

Calculator

1)Type the starting value P0 i.e. 10 Enter

2)Type 1.05Ans Enter

3) Keep pressing Enter to generate the pattern

Types of recurrence relations

What do the following recurrence relations do?

1) Un = 1.10Un–1

Increases the previous U value by 10% each time

2) Un = 2Un–1

Doubles the previous U value each time

3) Un = 0.05Un–1

Finds 5% of the previous U value

4) Un = 1.05Un–1–10

Increases the previous U value by 5% and subtracts 10 each time

More complex recurrence relations.

1) Paying off credit card bills.

A car costs £3000 and the loan company charges 2% interest per month.

You pay £300 off per month. How long does it take to repay?

At the end of month 1 the loan has increased by 2%.

Loan = 1.023000 = £3060

But £300 is paid off so £2760 is owed at the end of month 1

At the end of month 2 the loan has increased by 2%.

Loan = 1.022760 = £2815.20

But £300 is paid off so £2515.20 is owed at the end of month 2

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

Multiply by 1.02 to add 2% interest

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

Subtract £300

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

Multiply by 1.02 to add 2% interest

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

Subtract £300

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

Multiply by 1.02 to add 2% interest

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

Subtract £300

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

So to find out how much is owed the recurrence relation is Un = 1.02Un–1 – 300

No. months value

amount owing before

payment

owing after

paymentmthly

payment0 3000 3000 3001 3060 27602 2815.2 2515.2 interest %3 2565.5 2265.5 24 2310.81 2010.815 2051.03 1751.036 1786.05 1486.057 1515.77 1215.778 1240.09 940.0879 958.889 658.889

10 672.067 372.06711 379.508 79.508312 81.0985 -218.9

Spreadsheet

Link

Calculator

1)Type the starting value P0 i.e. 10 Enter

2)Type 1.2 Ans – 300 Enter

3) Keep pressing Enter to generate the pattern

The monthly interest rate can be easily changed to see the effect of different rates on the payment period.

The graph on the below shows how the outstanding loan decreases.

Loan oustanding after n months

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