AP & GP Fundamentals

What are AP and GP sequences?

In HSC Mathematics Extension 1, a sequence is an ordered list of numbers $(a_n)_{n\ge 1}$. An arithmetic progression (AP) has a constant difference $d$ between consecutive terms. A geometric progression (GP) has a constant ratio $r$. Once you know $a$ and $d$ (or $r$), you can find any term or the sum of the first $n$ terms with a single formula — that is the core skill this chapter builds.

Sequences can also be defined recursively (each term from previous ones) or by a worded rule. Exams often mix these forms, so practise translating between them.

How sequences are specified

• Explicit: $a_n$ written directly in terms of $n$, e.g. $a_n = 3n-1$.
• Recursive: $b_1=2$, $b_{n+1}=b_n+5$.
• Contextual: terms generated from a process (pricing, interest, geometry).

Always identify which form you are given before choosing a formula.

Arithmetic sequences and series

An AP satisfies $a_{n+1}-a_n=d$. With first term $a$:

$$a_n = a + (n-1)d.$$

The sum of the first $n$ terms is

$$S_n = \frac{n}{2}\bigl(2a+(n-1)d\bigr)=\frac{n}{2}(a+\ell),$$

where $\ell$ is the $n$th term. Use the second form when you already know the last term.

Worked example 1 — AP nth term

An arithmetic sequence has first term $4$ and common difference $5$. Find $a_{30}$.

Substitute into $a_n = a+(n-1)d$:

$$a_{30}=4+29\cdot 5=149.$$

Trap: multiply $d$ by $(n-1)$, not by $n$.

Worked example 2 — AP sum

Find the sum of the first $40$ terms of $9,\ 13,\ 17,\ \ldots$

Here $a=9$, $d=4$, $n=40$. The last term is $\ell=9+39\cdot 4=165$, so

$$S_{40}=\frac{40}{2}(9+165)=3480.$$

Geometric sequences and series

A GP satisfies $\dfrac{a_{n+1}}{a_n}=r$ (terms nonzero). With first term $a$:

$$a_n = ar^{n-1}.$$

For $r\neq 1$, the finite sum is

$$S_n = a\frac{1-r^n}{1-r}=a\frac{r^n-1}{r-1}.$$

Choose the form that keeps signs clean for your value of $r$.

Worked example 3 — Finite GP sum

Find $1+\frac12+\frac14+\cdots+\frac1{128}$.

This is a GP with $a=1$, $r=\frac12$. The last term is $(\frac12)^7$, so there are $n=8$ terms:

$$S_8=\frac{1-(\frac12)^8}{1-\frac12}=2\left(1-\frac1{256}\right)=\frac{255}{128}.$$

Sigma and Pi notation (quick rules)

Sigma notation packages a sum: $\sum_{k=1}^{n} a_k$. Useful habits for Ext 1:

• Shift indices carefully: $\sum_{k=1}^{n} a_{k+1}$ is not the same as $\sum_{k=2}^{n+1} a_k$ until you rewrite limits.
• Pull out constants: $\sum c\,a_k = c\sum a_k$.
• Split sums: $\sum (a_k+b_k)=\sum a_k+\sum b_k$.

Pi notation $\prod$ multiplies instead of adding; you meet it less often in Sequences, but the same index discipline applies.

Common mistakes

• Using $n$ instead of $n-1$ in $a_n = a+(n-1)d$ or $a_n=ar^{n-1}$.
• Mixing up AP difference and GP ratio on the same question.
• Forgetting that the finite GP sum formula requires $r\neq 1$ (if $r=1$, then $S_n=na$).
• Miscounting terms when the last term is given as a power of $r$.

Practice (hints only)

1. Given $a_3=11$ and $a_{12}=47$ in an AP, find $a$ and $d$, then $S_{25}$. (Set up two equations in $a$ and $d$.)
2. A GP has first term $5$ and ratio $2$. How many terms sum to $2555$? (Solve $5(2^n-1)=2555$.)

Full write-ups: see the related blog posts and the PDF fundamentals section.

Where next

Once AP/GP fluency is solid, move on to infinite series and recurring decimals — the same GP formula with $|r|<1$ and $n\to\infty$.