Number and Algebra
AA HL — Number and Algebra (sequences & series; polynomials & rational functions; exponentials & logarithms; binomial theorem, including the HL extension to rational index)
This topic is the toolkit you keep reaching for all the way through the course: every sequence problem, every log equation, every binomial expansion comes back to spotting a pattern and writing it as an exact algebraic statement. Examiners love 'show that' questions here precisely because the answer is already given — they're testing whether your algebra is airtight, not whether you can guess the right number.
Overview: how the four subtopics fit together
Sequences and series give you the language for anything that grows or decays step by step. Polynomials and rational functions extend that to smooth continuous behaviour — roots, asymptotes, and the algebra linking a function's coefficients to its zeros. Exponentials and logarithms are two views of the same relationship: one measures growth, the other measures 'how long until'. The binomial theorem is efficient multiplication — and at HL it's pushed further, letting the index be negative or fractional, which is exactly the machinery behind first-order approximations in physics and finance.
- A geometric sequence and an exponential function are the same idea in disguise — one is discrete (bounce number 1, 2, 3…), the other continuous (any real ).
- Proof by induction (HL only) is the tool examiners use to make you justify a general formula for a sequence rather than just quoting it.
- The HL extended binomial series and logarithms both come with a validity/domain condition attached — losing that condition line is the most common single mark dropped across this whole topic.
The shape of the chapter
Command terms that decide your method here
| Command term | What it demands | AO | Mark-earning move |
|---|---|---|---|
| Show that | Derive the given result with full algebraic steps — the answer is already known, so every line must be justified | AO1/AO2 | Marks for method lines, not just matching the final expression; skipping steps loses marks even if you land on the right answer |
| Hence | Use the result you just found — a fresh, independent method scores zero even if correct | AO2 | Method mark withheld unless you visibly use the previous part |
| Determine / Find | Obtain a numerical or algebraic answer; working can be minimal | AO2 | Correct answer alone can earn full marks, unless 'show that' or 'justify' is attached |
| Prove | Give a complete logical argument valid for all cases, often by induction | AO2/AO3 | Missing base case or concluding sentence in an induction proof costs a mark even with perfect algebra |
| Deduce | Use a previous result or reasoning to reach a new conclusion without re-deriving from scratch | AO3 | Must explicitly reference the earlier result to earn the mark |
Key point
Overview