Modelling · Topic 6 of 9
Recurrence Relations
Theory
Recurrence relations describe how each term in a sequence relates to the ones that come before it. They are incredibly valuable for modelling and analysing processes that evolve over time, such as population growth or financial forecasting.
1. The Structure of the Model
- A recurrence relation is a formula which generates a sequence of numbers, where each value is worked out from the one before it.
- Write it in the words of the question, so that it says exactly what you do to one value to get the next:
Pests next week = Pests this week × 0.45 + 400
- Always state your starting value — the amount at the very beginning, before any change has happened.
- In class, and in some textbooks, you will sometimes see this written with symbols instead: , with a starting value . That means exactly the same thing. In Higher Applications the question is always set in a context and answered in a spreadsheet, so you will not be asked to write it in that form.
2. Recurrence Relations in Spreadsheets (Excel)
At this level, we use spreadsheet software to do the hard work of calculating recurrence relations:
- Setup: First, set up two columns — one for the time period (year, week or day) and one for the quantity you are modelling.
- Entering the Formula: Put your starting value in the first cell of the quantity column. In the cell below it, type a formula starting with an
=sign, clicking on the cell directly above so that the formula references it. - Generate the Sequence: Click and drag the formula down to automatically fill the column and generate your sequence of numbers.
3. Long-Term Effects and Graphs
- If you drag the spreadsheet formula down through many rows, you can see the long-term effect of the model. Sometimes, the numbers will eventually reach a limit and stabilise at a certain amount.
- You can easily visualise this long-term trend by highlighting your data in Excel, clicking 'Insert', and selecting an appropriate chart (like a scatter or line graph).
Worked examples
Example 1
Example 1: Setting up a Spreadsheet Model
A local town has a population of 8,500 people. Each year, the population grows by 3%. At the same time, 150 people leave the town each year to move to a nearby city.
(a) The starting population is in cell B2 of a spreadsheet. Write down the formula you would enter in cell B3 so that it can be filled down the column.
(b) Explain the steps required to calculate the population of the town over the next 5 years using a spreadsheet.
Solutions:
- (a) Growing by 3% multiplies the population by 1.03, then 150 leave:
Population next year = Population this year × 1.03 − 150
In the spreadsheet:=B2*1.03-150 - (b) First, set up two columns, one for the year and one for the population. In the year column, fill in the numbers 0 to 5. In the population column, enter the starting value of 8500 next to year 0. In the cell below it, type a formula to multiply the cell above by 1.03 and subtract 150. Finally, drag the formula down to fill the rows up to year 5.
Example 2
Example 2: Extracting the Multiplier
A farmer adds 25 kg of compost to a field every month. Over the course of a month, 18% of the compost is absorbed by the soil or washed away. The farmer starts with an empty field (0 kg of compost).
The starting amount is in cell B2 of a spreadsheet. Write down the formula you would enter in cell B3 to model the amount of compost in the field at the end of each month.
Solution: If 18% is lost, then 82% remains. Therefore, the multiplier is 0.82.
Compost next month = Compost this month × 0.82 + 25
In the spreadsheet: =B2*0.82+25, with the starting value of 0 kg in cell B2.
Example 3
Example 3: Finding the Long-Term Limit
Some trees in a local orchard are infested with pests. The trees are sprayed weekly with a pesticide that manufacturers claim will destroy 55% of all pests. However, between weekly sprayings, it is estimated that 400 new pests invade the trees. Initially, there are 1,500 pests.
(a) The number of pests before the first spraying is in cell B2 of a spreadsheet. Write down the formula you would enter in cell B3 so that it can be filled down the column.
(b) Describe how to use a spreadsheet to determine what happens to the number of pests in the long term, and state the expected outcome.
Solutions:
- (a) If 55% are destroyed, 45% survive:
Pests next week = Pests this week × 0.45 + 400
In the spreadsheet:=B2*0.45+400 - (b) Enter the starting value of 1500 in the spreadsheet and create a formula in the cell below to multiply the previous cell by 0.45 and add 400. Drag the formula down to generate a sequence of numbers. By looking at the extended list of numbers, we can see the long-term effect: the population stabilises and levels out at approximately 727 pests.