Year 12 Discrete Mathematics: Recursive sequences

Recursive sequences describe change step by step. Instead of giving a direct formula for the $n$ th term, they tell you how each term is generated from the previous one or from several earlier terms. This makes them a natural starting point for discrete mathematics, because the whole topic is about processes that evolve in stages rather than continuously. The key skill is to keep track of what each symbol means. If you lose sight of which term is current and which term is being produced next, the notation becomes confusing very quickly.

A recursive rule is a process statement. Read it as instructions for moving from one stage to the next, not as a static algebraic expression.

Subtopic 1: Generating terms from a recursive rule

The simplest questions ask you to generate early terms. Here the aim is accuracy and interpretation. You need the initial value and the update rule, and then you apply that rule in order. Each new term is built from the one before it, so skipping a step or using the wrong term number breaks the whole chain. These questions may look straightforward, but they build the habits needed for much more complex models.

Worked example 1

Problem: A sequence is defined by

u_{1} = 5

and

u_{n + 1} = u_{n} + 4

. Find the first five terms.

The first term is given directly: $u_{1} = 5$ .
Apply the rule once: $u_{2} = 5 + 4 = 9$ .
Apply it again: $u_{3} = 9 + 4 = 13$ .
Continue: $u_{4} = 17$ and $u_{5} = 21$ .

Answer: the first five terms are

5, 9, 13, 17, 21

Subtopic 2: Interpreting the type of growth

Not all recursive sequences add a fixed amount. Some multiply by a constant, some combine both multiplication and addition, and some depend on more than one earlier term. Interpreting the structure matters because it tells you whether the growth is arithmetic-like, geometric-like, or more complex. Once you know the pattern type, you can predict the behaviour of the sequence more intelligently.

Worked example 2

Problem: A sequence is defined by

u_{1} = 2

and

u_{n + 1} = 3 u_{n} - 1

. Find the first four terms and describe the change.

Start with $u_{1} = 2$ .
Then $u_{2} = 3 (2) - 1 = 5$ .
Next $u_{3} = 3 (5) - 1 = 14$ .
Then $u_{4} = 3 (14) - 1 = 41$ .
The sequence is not arithmetic because the increase is not constant.
It is also not purely geometric because the rule multiplies and then subtracts.

Answer: the first four terms are

2, 5, 14, 41

, and the growth is multiplicative with a constant adjustment.

Recursive thinking in applications

Recursive sequences are powerful because they match many real situations naturally. Population models, savings plans, repeated discounts, and stage-by-stage decisions can all be described recursively. This is why the meaning of the rule matters more than the symbols alone. If a question says "each month the balance is increased by interest and then reduced by a fee," the recursion should reflect that order. In discrete mathematics, the interpretation of the update is often the whole problem.

Worked example 3

Problem: A balance starts at

$ 100

. Each week it is multiplied by

1.1

and then reduced by

$ 5

. Write a recursion and find the balance after two updates.

Let $B_{1} = 100$ .
The recursive rule is $B_{n + 1} = 1.1 B_{n} - 5$ .
Now calculate: $B_{2} = 1.1 (100) - 5 = 105$ .
Then $B_{3} = 1.1 (105) - 5 = 110.5$ .

Answer: one suitable recursion is

B_{1} = 100

B_{n + 1} = 1.1 B_{n} - 5

, and after two updates the balance is

$ 110.50

Common traps

Using the wrong previous term when applying the rule.
Forgetting the initial value and trying to generate terms from the rule alone.
Misreading the order of operations in a contextual update rule.
Assuming every recursion represents arithmetic or geometric change.

Revision focus

A strong revision habit is to read the recursion aloud in words before calculating. Saying "the next term is three times the current term minus one" is much safer than staring at the symbols and hoping the structure is obvious. This habit also makes worded applications much easier to translate into notation.

It also helps to tabulate the first few terms carefully. Writing term numbers next to the values makes the recursive pattern visible and prevents confusion about which index you are currently updating. In discrete mathematics, clean indexing is part of the mathematics, not just a presentation detail.

Practice links

Recursive sequences