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

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 to real-world contexts (loans, salaries, depreciation).

Arithmetic sequences

A sequence with a constant common difference $d$ between consecutive terms.

$T_1 = a$, $T_n = a + (n - 1)d$.

Sum of first $n$ terms: $S_n = \dfrac{n}{2}[2a + (n - 1)d] = \dfrac{n}{2}(T_1 + T_n)$.

Geometric sequences

A sequence with a constant common ratio $r$ between consecutive terms.

$T_1 = a$, $T_n = a r^{n - 1}$.

Sum of first $n$ terms: $S_n = a \dfrac{r^n - 1}{r - 1}$ for $r \neq 1$.

Sum to infinity (for $|r| < 1$): $S_\infty = \dfrac{a}{1 - r}$.

Identifying sequence type

Compute consecutive differences and ratios:

• Constant differences: arithmetic.
• Constant ratios: geometric.
• Neither: other type (quadratic, recursive, etc.).

Applications

Loans and savings. Compound interest gives a geometric sequence with $r = 1 + i$ where $i$ is the periodic interest rate.

Depreciation. Straight-line depreciation is arithmetic; declining-balance depreciation is geometric.

Salary growth. Fixed-dollar raise: arithmetic. Percentage raise: geometric.

Population. Constant absolute increase: arithmetic. Constant percentage growth: geometric.

Sum to infinity (geometric series)

For $|r| < 1$, the infinite series converges:

$S_\infty = a + ar + ar^2 + \cdots = \dfrac{a}{1 - r}$.

A bouncing ball that returns $80$% of its previous height (so $r = 0.8$) covers a total of $\dfrac{h}{1 - 0.8} = 5h$ vertically before stopping.

Worked example

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

Arithmetic with $a = 5$, $d = 3$.

$S_{20} = \dfrac{20}{2}[2(5) + 19(3)] = 10 \cdot [10 + 57] = 10 \cdot 67 = 670$.

Common traps

Confusing arithmetic and geometric. Add or multiply? Check by computing two consecutive differences and ratios.

Off-by-one on $n - 1$ vs $n$. $T_n$ uses $n - 1$; $S_n$ uses $n$.

Using sum-to-infinity when $|r| \ge 1$. Diverges; formula does not apply.

Mixing total and individual term. A question asking for the year $10$ salary wants $T_{10}$, not $S_{10}$.

In one sentence

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