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# FM Recurrence Relation

## 2.3 Extended Matrix Recurrence Relations

Note: if you cannot remember how to use simple matrix recurrence relations, revise notes for 2.1 Simple Matrix Recurrence Relations.

### The Extended Matrix Recurrence Formula

• The simple matrix recurrence formula we have analysed previously (in 2.1 Simple Matrix Recurrence Relations and 2.2 Applications of Matrix Recurrence Relations) only allows the values for the next state to be calculated based on proportional values of the current state. In some systems, this is not enough to produce an accurate model. The extended matrix recurrence formula allows us to model systems which have fixed values added or removed between states.
• The general form for the extended recurrence formula is:

S_{n+1}=T S_{n}+B

Where S_n is the state matrix for the nth state, T is the transition matrix and B is a matrix containing constant values.

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## 2.2 Applications of Matrix Recurrence Relations

Note: if you cannot remember how to use a matrix recurrence relation, revise notes for 2.1 Simple Matrix Recurrence Relations.

### Transition Diagrams

• Transition diagrams provide a visual method for modelling how a system changes between states.
• They consist of a series of nodes (dots) representing the variables in the system, with arrows drawn between them showing the movement of individuals/objects in the system between variables (e.g. the percentage of people who go from playing soccer one month to playing baseball the next).
• By convention numerical values should be shown as percentages in transition diagrams.
Read More »2.2 Applications of Matrix Recurrence Relations

## 4.1 Modelling Compound Interest Systems with Regular Withdrawals

Note: if you cannot remember the recurrence relation formula for a compound interest system, revise notes for 2.2 Analysis of Compound Interest.

### Modelling using a Recurrence Relation

• A compound interest system with regular withdrawals describes a system which has a positive interest rate, and withdrawals (negative) made at the end of each compounding period.
• We can use the simple recursion formula to model this system:

A_{n+1}=d+(1+I) A_{n}

Where d<0 and I>0.

Read More »4.1 Modelling Compound Interest Systems with Regular Withdrawals

## 1.2 Forms of Depreciation

### Flat Rate Depreciation

• Flat rate depreciation occurs when the value of an asset decreases by a fixed amount every period (i.e. no interest occurs).
• Graphs of flat rate depreciation are linear.
• The recurrence relation formula describing flat rate depreciation is:

A_{n+1}=d+A_{n}

Note: as this is depreciation: d<0. Consequently, any graphs showing flat rate depreciation will have a negative slope.

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## 1.1 Recurrence Relations and Sequences

### Sequence

A list of numbers written down in succession.

Terms of a Sequence

The terms of a sequence can be identified by their term numbers.

The initial term or 0th term is A_{0}; The first term is A_{1};The second term is A_{2}.

A_{0}, A_{1}, A_{2}

The 𝑛th term is 𝑢; The term before the 𝑛th term is 𝑢; The term after the 𝑛th term is 𝑢.

A_{n-1}, A_{n}, A_{n+1}

Any letter, not just A, can represent the terms of a sequence.

### Recurrence Relations

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