Sequences and Series summary

Key learning points

• Arithmetic: $u_n=a+(n-1)d$; $S_n=\frac{n}{2}(2a+(n-1)d)$.
• Geometric: $u_n=ar^{n-1}$; $S_n=\frac{a(r^n-1)}{r-1}$ ($r\ne1$).
• Infinite geometric sum when $|r|<1$: $S_\infty=\frac{a}{1-r}$.
• Always state which formula you use.

Worked example 1 — Linear nth term

Sequence $5,8,11,14,\ldots$. $n$th term?

Working

Common difference 3.

Worked example 2 — Next term

$2,6,12,20,\ldots$ next term?

Working

Second differences constant.

Worked example 3 — Membership

Is $50$ in $4n+2$?

Working

Solve for positive integer $n$.

Worked example 4 — Arithmetic sum

Sum of the first 20 terms of $3,7,11,\ldots$.

Working

Here $a=3$, $d=4$, so $\frac{20}{2}(6+76)=820$.

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.