Sequences and Arithmetic Series Topic Review summary

Key learning points

• $u_n=a+(n-1)d$ for first term $a$ and common difference $d$.
• $S_n=\frac{n}{2}(2a+(n-1)d)=\frac{n}{2}(a+l)$ with last term $l$.
• Find $a$ and $d$ from two given terms; watch inclusive counting.
• Prove a sequence is arithmetic by showing $u_{n+1}-u_n$ is constant.
• Mixed questions may combine arithmetic sums with inequalities or growth wording.

Worked example 1 — nth term

5th term of $a=3$, $d=4$.

Working

$a+(n-1)d$.

Worked example 2 — Sum

Sum of first 20 terms, $a=2$, $d=3$.

Working

$\frac{n}{2}(2a+(n-1)d)$.

Worked example 3 — u_n=a+(n-1)d for first term a and common d

$u_n=a+(n-1)d$ for first term $a$ and common difference $d$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — S_n=\n{2}(2a+(n-1)d)=\n{2}(a+l) with last

$S_n=\\frac{n}{2}(2a+(n-1)d)=\\frac{n}{2}(a+l)$ with last term $l$.

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.