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 to growth and decay problems. Builds the algebraic fluency Year 12 Methods will require.

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.

Applications

Compound interest. Principal $P$ at rate $r$ per period compounded for $n$ periods: $A = P(1 + r)^n$. The amounts form a geometric sequence.

Exponential growth and decay. Population, radioactive decay, drug clearance.

Annuities. Regular payments at compound interest. Formulas based on geometric sums.

Worked examples

Arithmetic. First term 3, common difference 4. Find $T_{10}$ and $S_{10}$.

$T_{10} = 3 + 9 \times 4 = 39$.

$S_{10} = (10/2)(3 + 39) = 5 \times 42 = 210$.

Geometric. First term 2, common ratio 0.5. Find $S_\infty$.

$S_\infty = 2 / (1 - 0.5) = 4$.

Common errors

Off-by-one in $T_n$. Use $n - 1$ in the exponent or multiplier, not $n$.

Applying infinite formula when divergent. $|r| < 1$ required.

Sign errors in $(r - 1)$ vs $(1 - r)$. Use the form that matches; both are correct but make sure your sign convention is consistent.

In one sentence

Arithmetic sequences have constant common difference $d$ with $T_n = a + (n-1)d$ and $S_n = (n/2)[2a + (n-1)d]$; geometric sequences have constant common ratio $r$ with $T_n = ar^{n-1}$ and $S_n = a(r^n - 1)/(r-1)$, plus the infinite series $S_\infty = a/(1-r)$ for $|r| < 1$; both apply to compound interest, exponential growth and decay.