Recognise arithmetic sequences, use the recursive and explicit rules, and apply them to simple interest and linear depreciation.

What this dot point is asking

You must recognise arithmetic behaviour, use both the recursive and explicit forms, find any term, sum a run of terms, and apply the model to simple interest and straight-line depreciation.

The common difference

A sequence is arithmetic when each term differs from the one before by the same constant, the common difference $d$. You test for it by subtracting consecutive terms: if $t_2 - t_1 = t_3 - t_2 = \dots$, the sequence is arithmetic. A positive $d$ gives steady growth (a straight line rising), a negative $d$ steady decline.

The explicit rule jumps straight to any term without listing the ones before it, which is why it is preferred for the later terms. The sum formula adds a whole run of terms without listing them, useful for total interest or total repayment questions.

Where arithmetic sequences appear

The arithmetic model fits any quantity that changes by a fixed amount per period.

• Simple interest. Interest is a fixed amount each period because it is always calculated on the original principal, so the balance grows arithmetically.
• Flat-rate (straight-line) depreciation. An asset loses the same dollar amount each year, so its value falls arithmetically with a negative $d$.
• Regular savings with no interest. Adding a fixed deposit each period builds an arithmetic balance.

Choosing recursive or explicit

Use the recursive rule when generating a table term by term or when technology iterates for you. Use the explicit rule when asked for a single far-off term, such as the value after $30$ years, because it avoids listing every step. Use the sum formula for totals, such as how much is repaid over the life of a flat fortnightly plan.

When you are given two terms (for example $t_3$ and $t_8$) and asked for $a$ and $d$, write both as $t_n = a + (n-1)d$, subtract to eliminate $a$ and solve for $d$, then back-substitute for $a$. This two-equation method appears almost every year.

Reading an arithmetic model from a worded context

SCSA questions rarely use the word "arithmetic". You decide the model from the wording. Phrases such as "increases by $\$40$ each month", "loses $\$2400$ a year" or "adds $250$ litres per day" all signal a fixed step, so the quantity is arithmetic and you set $d$ to that step (negative if it falls). Phrases such as "increases by $5\%$ each year" signal a fixed multiplier, so the quantity is geometric instead, and you must switch models. Identifying the step versus the multiplier is the single decision that determines every later line of working, so do it before reaching for a formula.

Once the model is fixed, the three tools follow directly: the recursive rule for a table, the explicit rule $t_n = a + (n-1)d$ for a distant term, and the sum $S_n = \dfrac{n}{2}(2a + (n-1)d)$ for a running total such as total interest, total saved or total distance.