VICGeneral MathematicsSyllabus dot point

How do first-order recurrence relations generate arithmetic and geometric sequences, and how are their terms and sums found?

Use a first-order linear recurrence relation to generate a sequence, recognise arithmetic and geometric sequences, find the nth term with an explicit rule, and compute the sum of an arithmetic or geometric series

A focused answer to the VCE General Mathematics Unit 3 Recursion and financial modelling key-knowledge point on sequences. First-order recurrence relations, arithmetic and geometric rules, explicit nth-term formulas, and the sums of arithmetic and geometric series.

Generated by Claude Opus 4.76 min answerUpdated 2026-05-29

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
Arithmetic sequences
Geometric sequences
Generating a sequence from a recurrence
Linking the type to the context
Why this matters for the exams

What this dot point is asking

VCAA wants you to generate and analyse sequences using a first-order linear recurrence relation, a rule that gives each term from the previous one together with a starting value. You must recognise when a recurrence produces an arithmetic sequence (constant difference) or a geometric sequence (constant ratio), switch between the recursive and explicit forms, and find the sum of a series. Recursion is the engine behind all the financial models in Unit 3, so getting fluent here pays off across the whole core.

Arithmetic sequences

An arithmetic sequence adds a constant common difference $d$ each step. Its recurrence is $t_{n+1} = t_n + d$ with first term $t_1 = a$ . The explicit rule for the nth term is

t_n = a + (n - 1)d.

The sum of the first $n$ terms is

S_n = \frac{n}{2}\big(2a + (n-1)d\big) = \frac{n}{2}\big(a + t_n\big).

Arithmetic growth is linear: plot the terms and they sit on a straight line. Simple interest and flat (straight-line) depreciation are arithmetic.

Geometric sequences

A geometric sequence multiplies by a constant common ratio $R$ each step. Its recurrence is $t_{n+1} = R\,t_n$ with first term $t_1 = a$ . The explicit rule is

t_n = a\,R^{\,n-1}.

The sum of the first $n$ terms (for $R \ne 1$ ) is

S_n = \frac{a(R^n - 1)}{R - 1}.

Geometric growth is exponential: the terms curve upward when $R > 1$ and decay toward zero when $0 < R < 1$ . Compound interest and reducing-balance (unit-cost) depreciation are geometric.

Generating a sequence from a recurrence

Given $t_{n+1} = R\,t_n + d$ and $t_1$ , you can list terms by hand or on the calculator. When $d = 0$ the sequence is purely geometric; when $R = 1$ it is purely arithmetic. The mixed case (both $R \ne 1$ and $d \ne 0$ ) models reducing-balance loans and annuities, which build directly on this dot point.

Worked example

Arithmetic and geometric calculations

An arithmetic sequence is defined by $t_{n+1} = t_n + 4$ with $t_1 = 7$ .

Explicit rule: $t_n = 7 + (n-1)\times 4 = 4n + 3$ .

The 10th term: $t_{10} = 4 \times 10 + 3 = 43$ .

Sum of the first 10 terms: $S_{10} = \dfrac{10}{2}\big(7 + 43\big) = 5 \times 50 = 250$ .

A geometric sequence is defined by $t_{n+1} = 3\,t_n$ with $t_1 = 2$ .

Explicit rule: $t_n = 2 \times 3^{\,n-1}$ .

The 5th term: $t_5 = 2 \times 3^{4} = 2 \times 81 = 162$ .

Sum of the first 5 terms: $S_5 = \dfrac{2(3^5 - 1)}{3 - 1} = \dfrac{2(243 - 1)}{2} = 242$ .

Linking the type to the context

The reason this dot point matters is that every financial model in Unit 3 is one of these two patterns plus a constant.

Simple interest adds the same dollar amount each period: arithmetic.
Compound interest multiplies by a fixed factor each period: geometric.
A reducing-balance loan multiplies by the growth factor and then subtracts a repayment: a mixed first-order recurrence.

Recognising the underlying sequence tells you instantly whether growth is linear or exponential and which formula to reach for.

Common trap

Off-by-one in the nth-term rule: The explicit rules use $n - 1$ , not $n$ . Forgetting this makes every term one step out. Check by substituting $n = 1$ ; it must return the first term.
Using the wrong sum formula: The arithmetic and geometric sum formulas are different. Identify the sequence type first by testing for a common difference or a common ratio.
Mixing recursive and explicit forms: $t_{n+1} = t_n + d$ describes the step; $t_n = a + (n-1)d$ gives any term directly. Use the explicit form when asked for a specific far-out term.
Dividing by zero in the geometric sum: The formula $\dfrac{a(R^n - 1)}{R - 1}$ requires $R \ne 1$ . If $R = 1$ the sequence is constant and the sum is simply $na$ .

Why this matters for the exams

Recurrence questions appear on both exams and are quick marks once the pattern is recognised. Exam 1 often asks you to identify the sequence type or apply an explicit rule by hand, while Exam 2 may ask you to run the recurrence on a calculator to find a later term or a sum. The mixed recurrence introduced here is exactly the model used for loans, investments and annuities in the next dot point.

Exam-style practice questions

Practice questions written in the style of VCAA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

2023 VCAA1 marksA sequence of numbers is generated by the recurrence relation T0 = 5, T(n+1) = -T(n). The value of T2 is A. -10 B. -5 C. 0 D. 5 E. 10

Show worked answer →

Apply the rule one step at a time, starting from T0 = 5.

T1 = -T0 = -5.

T2 = -T1 = -(-5) = 5.

So the answer is D. The rule simply flips the sign each step, producing 5, -5, 5, -5, so the even-numbered terms equal the starting value.

2025 VCAA1 marksA recurrence relation is of the form u0 = a, u(n+1) = R u(n) + d. If a > 0, R = 0.5 and d = 0, the sequence generated will be A. arithmetic and increasing. B. arithmetic and decreasing. C. geometric and increasing. D. geometric and decreasing.

Show worked answer →

With d = 0 the rule becomes u(n+1) = 0.5 x u(n). Each term is a fixed multiple (0.5) of the previous term, which is the definition of a geometric sequence (a common ratio, not a common difference), so it is not arithmetic.

Since a > 0 and the common ratio is 0.5 (between 0 and 1), each term is half the size of the one before, so the terms get smaller and the sequence is decreasing.

Geometric and decreasing matches option D.