Modelling with Sequences and Series

Understanding Modelling

  • Modelling is a process used in mathematics to describe real-world situations and phenomena using mathematical language or structures.
  • The concept of modelling is crucial as it gives practical context to abstract mathematical ideas and provides a way to predict or analyse real-world systems.

Modelling with Sequences

  • A sequence can be used to model a scenario where a pattern or rule is continually repeated.
  • For example, the growth of a population, where each term represents the population size after a certain time period, can be modelled using a sequence.
  • Making predictions about future situations (like predicting a future term in a series) is often a key part of modelling with sequences.

Modelling with Arithmetic Sequences

  • Arithmetic sequences can be used to model situations where there is a constant rate of change.
  • For instance, the appreciation or depreciation in the value of an asset over time (car value depreciation, for example) can be typically modelled using an arithmetic sequence.
  • The nth term formula a + (n - 1)d should be used to predict future values in the model, where a is the initial value and d is the rate of change.

Modelling with Geometric Sequences

  • Geometric sequences are often used to model situations that involve exponential growth or decay.
  • Examples include economic situations involving compound interest or rates of inflation, as well as biological phenomena such as the growth of bacteria in a culture.
  • Predictions in the model are made using the nth term formula ar^(n-1), where a is the initial value and r is the common ratio.

Modelling with Fibonacci Sequences

  • Fibonacci sequences, although less common, have been used in modelling biological growth processes, particularly those that involve replication or reproduction.
  • For example, a Fibonacci sequence can be used to predict the growth of a bee population or the arrangement of seeds in a sunflower head.

Investigating Models

  • Once a sequence or series has been used to model a situation, its validity should be tested - for example, by checking the model’s predictions against collected data.
  • Models may often need to be refined, which includes tweaking parameters, using a different kind of sequence or adjusting the model’s assumptions.
  • The final mark of a good model is how well it predicts or describes the scenario it was designed to represent.