Lesson Plan

• Describe the condition for convergence of a geometric series (|r| < 1).
• Derive and apply the sum‑to‑infinity formula \(S_{\infty}=a/(1-r)\) for a convergent GP.
• Identify the first term and common ratio in a given series and compute \(S_{\infty}\) or finite sums.
• Analyse common misconceptions and justify why certain series diverge.

• Projector and screen
• Whiteboard and markers
• Student worksheets (GP recap, practice questions)
• Scientific calculators
• Handout summarising key GP formulas
• Exit‑ticket slips

Begin with a quick visual of a shrinking geometric sequence approaching a line to spark curiosity. Review the definition of a GP and the finite‑sum formula, checking that students can identify \(a\) and \(r\). State that by the end of the lesson they will be able to decide if a series converges and find its infinite sum.

1. Do‑now (5'): Students complete a short recap worksheet on GP terms and the finite‑sum formula.
2. Mini‑lecture (10'): Present the convergence condition \(|r|<1\) and derive \(S_{\infty}=a/(1-r)\) using the limit of \(S_n\).
3. Guided example (10'): Work through the provided example (5 + 2 + …); students fill in steps on their handout.
4. Collaborative practice (15'): In pairs, solve practice questions 1–3, with teacher circulating to probe reasoning.
5. Class discussion (10'): Review answers, highlight common mistakes from the notes (e.g., using the formula when \(|r|\ge1\)).
6. Exit ticket (5'): Each student writes the convergence condition and computes \(S_{\infty}\) for a new GP (e.g., \(a=4, r=‑0.6\)).

Summarise the key steps: check \(|r|<1\), then apply \(S_{\infty}=a/(1-r)\). Collect exit tickets to gauge understanding. For homework, students finish the remaining practice question and create their own convergent GP, stating the sum to infinity.