# Modelling

# 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.