QCE General Mathematics - Unit 3 - Growth and decay in sequences
Arithmetic Sequences | QCE General Mathematics
Learn recursion, tables, graphs and nth-term rules for arithmetic sequences in QCE General Mathematics.
Updated 2026-05-18 - 5 min read
QCAA official coverage - General Mathematics 2025 v1.3
Exact syllabus points covered
- Use recursion to generate an arithmetic sequence.
- Display the terms of an arithmetic sequence in both tabular and graphical form and demonstrate that arithmetic sequences can be used to model linear growth and decay in discrete situations.
- Use the rule for the $n$th term of an arithmetic sequence: $t_n=t_1+(n-1)d$, where $t_n$ is the $n$th term, $t_1$ is the first term, $n$ is the term number and $d$ is the common difference.
- Use arithmetic sequences to model and analyse practical situations involving linear growth or decay, e.g. analysing a simple interest loan or investment, calculating a taxi fare based on the flag fall and the charge per kilometre, calculating the value of an item using the straight-line method of depreciation.
An arithmetic sequence changes by adding the same amount each time. That amount is called the common difference. Arithmetic sequences model discrete linear growth and decay, such as a taxi fare increasing by a fixed charge per kilometre or an asset losing the same dollar amount each year under straight-line depreciation.
Original Sylligence diagram for general arithmetic geometric sequences.
Recursive form
Recursive form tells you how to get the next term from the current term:
$ t_{n+1}=t_n+d $
where $d$ is the common difference.
If $t_1=18$ and $d=7$, the sequence is:
$ 18,\ 25,\ 32,\ 39,\ldots $
The nth-term rule
The direct rule is:
$ t_n=t_1+(n-1)d $
This is faster when you need a later term because you do not have to list every term before it.
Worked example
Common traps
State what $n=1$ represents in context. In finance and depreciation questions, "year 1" and "after 1 year" may not mean the same time point.
Checking for arithmetic behaviour
To test whether a sequence is arithmetic, subtract consecutive terms:
$ t_2-t_1,\quad t_3-t_2,\quad t_4-t_3,\ldots $
If the difference is constant, the sequence is arithmetic. The common difference can be positive, negative or zero.
| Sequence | Common difference | Interpretation | |---|---:|---| | $5,9,13,17,\ldots$ | $4$ | linear growth | | $80,72,64,56,\ldots$ | $-8$ | linear decay | | $11,11,11,\ldots$ | $0$ | no change |
Tables, graphs and simple interest
Arithmetic sequences appear as equally spaced points on a straight line when term number is graphed against term value. They are discrete because only whole-number term positions make sense.
Simple interest is an arithmetic-sequence model because the same amount of interest is added each period:
$ A_n=P+nI $
where $I$ is the interest earned per period. This differs from compound interest, where the increase changes because interest is calculated on a changing balance.
Depth: identifying arithmetic structure
An arithmetic sequence has a constant difference between consecutive terms:
$ d=t_{n+1}-t_n $
The difference can be positive, negative or zero.
| Sequence | Difference | Behaviour | |---|---:|---| | $7, 11, 15, 19,\ldots$ | $4$ | increasing linearly | | $50, 42, 34, 26,\ldots$ | $-8$ | decreasing linearly | | $12, 12, 12, 12,\ldots$ | $0$ | constant |
If the differences are not constant, the sequence is not arithmetic.
Rule forms
The recursive rule is:
$ t_{n+1}=t_n+d $
The explicit rule is:
$ t_n=t_1+(n-1)d $
The explicit rule is better when you need a far-away term. The recursive rule is better when building a table step by step.
Arithmetic sequences and simple interest
Simple interest creates arithmetic growth because the same amount of interest is added each period.
If $P$ dollars earns simple interest at rate $r$ per period, then the balance after $n$ periods can be modelled as:
$ A_n=P+nPr $
depending on whether $n$ counts completed interest periods. In sequence questions, check whether $t_1$ is the starting amount or the amount after one period.
Graph interpretation
When arithmetic sequence terms are plotted against $n$, the points lie on a straight line. The common difference is the slope between consecutive points. This is why arithmetic growth is also called linear growth.
Depth: arithmetic sums
Some sequence contexts ask for a total, not just a single term. The sum of the first $n$ terms of an arithmetic sequence is:
$ S_n=\frac{n}{2}(t_1+t_n) $
or:
$ S_n=\frac{n}{2}(2t_1+(n-1)d) $
Check whether the question asks for the amount in a particular period or the accumulated total over many periods.
A useful final check is units: if the terms represent dollars per week, the sum represents total dollars, not dollars per week.