4.10: Recursive Equations in Spreadsheets
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How do we efficiently make long range predictions? Or more generally, how do we make a prediction more than one or two time periods into the future? Direct computations using a recursive equation are very tedious. A good approach — useful for any recursive relationship — is to use a spreadsheet such as Excel.
Example 39 — Logistic Growth in a Spreadsheet
Suppose a new high-tech device is introduced into a society (like a smart phone or electric vehicle). The rate of adoption is initially slow, then appears exponential, but eventually levels off as the market saturates. This may be modeled by a logistic equation.
Consider the following hypothetical data showing the percent of the population which has adopted a certain new technology:
| Year (n) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Adoption % | 8 | 13 | 22 | 33 | 45 | 58 | 68 | 75 | 81 | 83 | 82 |
The following logistic growth model has been determined:
To use a spreadsheet to generate predicted values:
- Enter the original data in columns A (Year) and B (Observation).
- In cell C2, enter the initial value:
8 - In cell C3, enter the recursive formula:
= C2 + 0.66*(1 – C2/83)*C2 - Copy the formula from C3 and paste it into cells C4 through C12.
As long as you used cell references in the formula, Excel will automatically continue the recursive pattern. The model predicts values very close, but not identical, to the observations:
| Year | Observation | Prediction |
|---|---|---|
| 0 | 8 | 8.0 |
| 1 | 13 | 12.8 |
| 2 | 22 | 19.9 |
| 3 | 33 | 29.9 |
| 4 | 45 | 42.5 |
| 5 | 58 | 56.2 |
| 6 | 68 | 68.2 |
| 7 | 75 | 76.2 |
| 8 | 81 | 80.3 |
| 9 | 83 | 82.0 |
| 10 | 82 | 82.7 |
Example 40 — Modeling Disease Spread
Assume a single individual with a simulated infectious disease (simulitis) enters a population of size 500. Absent any constraints, each infected individual would cause one new infection per week. The logistic recursive equation is:
The initial number of infections is P0 = 1. In a spreadsheet:
- Enter column headings and the initial value 1 in cell B2.
- In cell B3, enter:
= B2 + 1*(1 – B2/500)*B2 - Copy the formula down for 12 rows to model 12 weeks.
- Set cells to display values to the nearest whole number.
Try it Now
A field currently contains 20 mint plants. Absent constraints, the number of plants would increase by 70% each year, but the field can only support a maximum population of 300 plants. Set up a spreadsheet using the logistic model to predict the population for the next 10 years. The recursive equation is:
This section is remixed from Quantitative Reasoning (Lachniet et al., 2026), §2.8, licensed CC-BY-SA.