Arithmetic and geometric sequences and series, including the general term formulas, sum formulas, and applications to growth and decay problems

What this dot point is asking

QCAA wants Year 11 students to define arithmetic and geometric sequences and series, compute terms and sums, and apply them to growth and decay problems. This builds the algebraic fluency Year 12 Methods relies on, and connects directly to compound interest and exponential models.

What a sequence is

A sequence is an ordered list of numbers generated by a rule, with $T_n$ denoting the $n$th term. The two families studied here are defined by how each term relates to the previous one: arithmetic sequences add a fixed amount, geometric sequences multiply by a fixed factor. Identifying which family a sequence belongs to, by checking whether consecutive differences or consecutive ratios are constant, is the first decision in every problem and determines which formulas apply.

Arithmetic sequences

A sequence where each term differs from the previous by a constant $d$ (common difference).

General term. $T_n = a + (n - 1) d$ where $a$ is the first term.

Sum of first $n$ terms. $S_n = (n/2)[2a + (n - 1) d]$ or equivalently $S_n = (n/2)(a + T_n)$.

Geometric sequences

A sequence where each term is the previous multiplied by a constant $r$ (common ratio).

General term. $T_n = a r^{n-1}$.

Sum of first $n$ terms. $S_n = a(r^n - 1)/(r - 1)$ for $r \neq 1$. (If $r = 1$, $S_n = na$.)

Infinite geometric series (for $|r| < 1$): $S_\infty = a / (1 - r)$.

The infinite formula requires convergence ($|r| < 1$); otherwise the series diverges.

Recurrence and explicit forms

A sequence can be described either recursively, giving each term in terms of the previous one (for example $T_{n+1} = T_n + d$ for arithmetic, or $T_{n+1} = rT_n$ for geometric), or explicitly, giving $T_n$ directly as a formula in $n$. The explicit form is what you need to find a distant term without listing all the earlier ones, and deriving the explicit form from a described recurrence is a common first step in modelling questions.

Applications

Compound interest

A principal $P$ at rate $r$ per period compounded for $n$ periods grows to $A = P(1 + r)^n$, and the successive balances form a geometric sequence with common ratio $1 + r$. This is the most direct application of geometric sequences in the course.

Exponential growth and decay

Population growth, radioactive decay and drug clearance all change by a constant factor per period, making them geometric and linking directly to the exponential functions studied alongside.

Annuities and repeated payments

A stream of regular payments earning compound interest sums as a geometric series, so the sum formula $S_n = a\dfrac{r^n - 1}{r - 1}$ values the whole stream. This idea underpins savings plans and loan repayments met later in the course.

Sequence versus series

A sequence is an ordered list of terms ($T_1, T_2, T_3, \ldots$); a series is the sum of its terms ($S_n = T_1 + T_2 + \cdots + T_n$). Questions often hinge on this distinction: "the fifth term" is $T_5$, a single value, while "the sum of the first five terms" is $S_5$. The general-term formulas describe the sequence, and the sum formulas describe the series. Reading which one a question wants is the first step to avoiding an avoidable error.