Arithmetic and geometric sequences (QCE Math Methods Unit 1)

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 $T_1 = a$,

The sum of the first $n$ terms is

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 $T_1 = a$,

The sum of the first $n$ terms is

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

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.

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 $|r| < 1$.