Geometric Series summary

Key learning points

• $n$th term $ar^{n-1}$; $S_n=\frac{a(1-r^n)}{1-r}$ (equivalent forms OK).
• Infinite sum only when $|r|<1$: $S_\infty=\frac{a}{1-r}$.
• Identify $a$ and $r$ carefully from the wording.
• Recurring decimals are infinite geometrics in disguise.

Worked example 1 — Infinite sum

Sum to infinity of $5+\frac{5}{2}+\frac{5}{4}+\cdots$.

Working

$a=5$, $r=\frac{1}{2}$, so $\frac{5}{1-1/2}=10$.

Worked example 2 — th term ; (equivalent forms OK)

Use this method from the Geometric Series sheet: $n$th term $ar^{n-1}$; $S_n=\frac{a(1-r^n)}{1-r}$ (equivalent forms OK).

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 3 — Infinite sum only when

Use this method from the Geometric Series sheet: Infinite sum only when $|r|<1$: $S_\infty=\frac{a}{1-r}$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 4 — Identify and carefully from the wording

Use this method from the Geometric Series sheet: Identify $a$ and $r$ carefully from the wording.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 5 — Recurring decimals are infinite geometrics in disguise

Use this method from the Geometric Series sheet: Recurring decimals are infinite geometrics in disguise.

Working

Each key learning point matches a question type on the printable worksheet.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.