What is off-by-one on the exponent?

The nth term uses n - 1 (T_n = ar^n-1), while the sum uses n (r^n in the numerator). Mixing these is a frequent slip.

QCAA QCAA 2022-style practice: Paper 2 (complex familiar). A worker is paid \45\,000in year1with a4\%annual raise. (a) Determine the salary in year10. (b) Determine the total earned over the first10$ years.

Geometric sequence with a = 45\,000, r = 1.04. (a) T_10 = 45\,000 × (1.04)^9 = 45\,000 × 1.4233 ≈ \64\,049.$ (b) S_10 = a\,r^10 - 1r - 1 = 45\,000 × 1.04^10 - 10.04 = 45\,000 × 12.0061 ≈ \540\,275.$ Markers reward identifying a and r, the correct formula choice, and answers in dollars.

QLDMath MethodsSyllabus dot point

How are arithmetic and geometric sequences analysed?

Define arithmetic and geometric sequences, find the $n$ th term and the sum of the first $n$ terms, and apply to real-world contexts

A focused answer to the QCE Math Methods Unit 1 dot point on sequences. States the $n$ th-term and sum formulas for arithmetic and geometric sequences, distinguishes the two types, and works the standard QCAA applications to salary growth, depreciation and convergent series.

Generated by Claude Opus 4.88 min answerUpdated 2026-06-02

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

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What this dot point is asking
Arithmetic sequences
Geometric sequences
Identifying the sequence type
Applications
Sum to infinity
In one sentence

What this dot point is asking

QCAA wants you to define and analyse arithmetic and geometric sequences, find the $n$ th term and the sum of the first $n$ terms, and apply both to real-world contexts such as loans, salaries, depreciation and population change. Sequences appear in Unit 1 problem-solving and modelling tasks and recur in the financial-mathematics content, so the formulas and the decision of which type applies are foundational skills.

Arithmetic sequences

An arithmetic sequence has a constant common difference $d$ between consecutive terms, so each term is the previous one plus $d$ . With first term $T_1 = a$ ,

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}(T_1 + T_n),

the second form being the average of the first and last term multiplied by how many terms there are. The common difference can be found from any two terms by $d = \dfrac{T_m - T_k}{m - k}$ .

Geometric sequences

A geometric sequence has a constant common ratio $r$ between consecutive terms, so each term is the previous one multiplied by $r$ . With first term $T_1 = a$ ,

T_n = a\,r^{n - 1}.

The sum of the first $n$ terms is

S_n = a\,\frac{r^n - 1}{r - 1}, \qquad r \neq 1.

When $|r| < 1$ the terms shrink toward zero and the infinite series converges:

S_\infty = \frac{a}{1 - r}.

The common ratio is found by dividing any term by the one before it, $r = \dfrac{T_{n+1}}{T_n}$ .

Identifying the sequence type

The first decision in any sequence problem is which type you have. Compute the differences and ratios of consecutive terms: constant differences signal arithmetic, constant ratios signal geometric, and if neither is constant the sequence is some other type (for example quadratic or recursively defined). Making this check explicitly avoids applying the wrong formula, which is the single most common error.

Applications

The two sequence types model different real-world growth. A fixed-dollar pay rise, straight-line depreciation, and a constant absolute population increase are all arithmetic, because the same amount is added each period. A percentage pay rise, compound interest, declining-balance depreciation, and constant-percentage population growth are all geometric, because the quantity is multiplied by the same factor each period. For compound interest at periodic rate $i$ , the balance forms a geometric sequence with $r = 1 + i$ .

Sum to infinity

For a convergent geometric series ( $|r| < 1$ ), the partial sums approach the limit $S_\infty = \dfrac{a}{1 - r}$ . This models a total made up of ever-smaller contributions. A ball that rebounds to $80\%$ of its previous height ( $r = 0.8$ ) travels a total downward distance of $\dfrac{h}{1 - 0.8} = 5h$ before coming to rest. The formula fails when $|r| \geq 1$ because the terms do not shrink and the series diverges.

Sum of a geometric series in an application

Set up the sequence

A patient takes a $200$ mg dose of a drug each day, and each day the body eliminates $25\%$ of whatever is present, leaving $75\%$ . The long-term amount just after a dose is the sum of all remaining contributions, a geometric series with $a = 200$ and $r = 0.75$ .

Apply the sum-to-infinity formula

Since $|r| = 0.75 < 1$ , the series converges:

S_\infty = \frac{a}{1 - r} = \frac{200}{1 - 0.75} = \frac{200}{0.25} = 800 \text{ mg}.

Interpret the result

The drug level stabilises at about $800$ mg just after each dose. Markers reward recognising the geometric structure, checking $|r| < 1$ before using $S_\infty$ , and the correct steady-state value with units.

Sum of an arithmetic sequence

Find the sum of the first $20$ terms of $5, 8, 11, 14, \ldots$

Step 1: Identify the sequence type and its parameters

Check whether consecutive terms differ by a constant amount. Here $8 - 5 = 3$ , $11 - 8 = 3$ , $14 - 11 = 3$ , so the sequence is arithmetic with first term $a = 5$ and common difference $d = 3$ .

Step 2: Choose the correct sum formula

Because we know $a$ , $d$ , and $n$ , the most direct formula is:

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

Substitute $n = 20$ , $a = 5$ , $d = 3$ :

S_{20} = \frac{20}{2}\big[2(5) + 19(3)\big]

Step 3: Evaluate inside the bracket first, then multiply

Work from the inside out to avoid errors:

2(5) = 10, \qquad 19(3) = 57, \qquad 10 + 57 = 67

S_{20} = 10 \times 67 = 670

Final answer: The sum of the first $20$ terms is $670$ .

In one sentence

Arithmetic sequences have common difference $d$ with $T_n = a + (n - 1)d$ and $S_n = \tfrac{n}{2}(T_1 + T_n)$ , while geometric sequences have common ratio $r$ with $T_n = ar^{n-1}$ , $S_n = a\dfrac{r^n - 1}{r - 1}$ , and $S_\infty = \dfrac{a}{1 - r}$ for $|r| < 1$ .

Exam-style practice questions

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

QCAA 20224 marksPaper 2 (complex familiar). A worker is paid

\

45\,000

in year

with a

4\%

annual raise. (a) Determine the salary in year

. (b) Determine the total earned over the first

10$ years.

Show worked answer →

Geometric sequence with $a = 45\,000$ , $r = 1.04$ .

(a) $T_{10} = 45\,000 \times (1.04)^9 = 45\,000 \times 1.4233 \approx \$ 64,049.$

(b) $S_{10} = a\,\dfrac{r^{10} - 1}{r - 1} = 45\,000 \times \dfrac{1.04^{10} - 1}{0.04} = 45\,000 \times 12.0061 \approx \$ 540,275.$

Markers reward identifying $a$ and $r$ , the correct formula choice, and answers in dollars.

QCAA 20233 marksPaper 1 (technique). The third term of an arithmetic sequence is

17

and the seventh term is

33

. (a) Determine the first term and common difference. (b) Determine the sum of the first

20

terms.

Show worked answer →

(a) $T_3 = a + 2d = 17$ and $T_7 = a + 6d = 33$ . Subtracting, $4d = 16$ , so $d = 4$ and $a = 17 - 2(4) = 9.$

(b) $S_{20} = \dfrac{20}{2}[2(9) + 19(4)] = 10[18 + 76] = 10 \times 94 = 940.$

Markers reward setting up two equations, solving for $a$ and $d$ , and the sum-formula evaluation.