Geometric Series Topic Review summary

Key learning points

• $u_n=ar^{n-1}$; identify $a$ and $r$ carefully.
• $S_n=\frac{a(1-r^n)}{1-r}$ (equivalent forms OK).
• Infinite sum only when $|r|<1$: $S_\infty=\frac{a}{1-r}$.
• Given two terms, form equations in $a$ and $r$; watch the sign of $r$.
• Recurring decimals are infinite geometrics in disguise.

Worked example 1 — Infinite sum

$5+\frac{5}{2}+\frac{5}{4}+\cdots$.

Working

$a=5$, $r=\frac12$.

Worked example 2 — Ratio

5th term 405, 9th 32805 (positive). Find $r$.

Working

$r^4=81$.

Worked example 3 — u_n=ar^{n-1}; identify a and r carefully

$u_n=ar^{n-1}$; identify $a$ and $r$ carefully.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — S_n=\a(1-r^n){1-r} (equivalent forms OK)

$S_n=\\frac{a(1-r^n)}{1-r}$ (equivalent forms OK).

Working

Attempt without notes, then check the answers PDF.

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.