Knowledge organisers

Number and algebra: IB Maths AA SL knowledge organiser

Everything to know about number and algebra on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Arithmetic sequence: Each term is the previous term plus the same number d, the common difference.
Geometric sequence: Each term is the previous term times the same number r, the common ratio.
Logarithm: log_a b is the power you raise a to in order to get b, so log_a b = x means aˣ = b.
Binomial coefficient: ⁿCᵣ counts the ways to choose r items from n; these numbers are the coefficients in the expansion of (a + b)ⁿ.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Arithmetic sequenceIn the formula booklet	\(u_n=u_1+(n-1)d\)
Arithmetic seriesIn the formula booklet	\(S_n=\tfrac n2\big(2u_1+(n-1)d\big)=\tfrac n2(u_1+u_n)\)
Geometric sequenceIn the formula booklet	\(u_n=u_1r^{n-1}\)
Geometric series, \(r\ne1\)In the formula booklet	\(S_n=\frac{u_1(r^n-1)}{r-1}=\frac{u_1(1-r^n)}{1-r}\)
Sum to infinity, \(\|r\|<1\)In the formula booklet	\(S_\infty=\frac{u_1}{1-r}\)
Compound interest (\(k\) times a year, \(n\) years, \(r\)%)In the formula booklet	\(FV=PV\left(1+\tfrac{r}{100k}\right)^{kn}\)
Exponents & logsIn the formula booklet	\(a^x=b\iff x=\log_ab,\) \(a^x=e^{x\ln a},\) \(\log_aa^x=x=a^{\log_ax}\)
Laws of logsIn the formula booklet	\(\log_axy=\log_ax+\log_ay,\) \(\log_a\tfrac xy=\log_ax-\log_ay,\) \(\log_ax^m=m\log_ax\)
Change of baseIn the formula booklet	\(\log_ba=\frac{\log_ca}{\log_cb}\)
Special values	\(\log_a1=0,\) \(\log_aa=1,\) \(\ln e=1,\) \(e^{\ln x}=x\)
Binomial theorem, \(n\in\mathbb N\)In the formula booklet	\((a+b)^n=a^n+\tbinom n1a^{n-1}b+\cdots\) \({}+\tbinom nra^{n-r}b^r+\cdots+b^n\)
Binomial coefficientIn the formula booklet	\(\binom nr={}^nC_r=\frac{n!}{r!\,(n-r)!}\)
General term	\(T_{r+1}=\binom nra^{n-r}b^r\)

Worked example

A geometric sequence has first term 5 and common ratio 3. Find the sum of the first 6 terms.

S₆ = u₁(rⁿ − 1)/(r − 1) with u₁ = 5, r = 3, n = 6
S₆ = 5(3⁶ − 1)/(3 − 1) = 5 × 728 / 2

Answer: S₆ = 1820

Common mistakes

Sequences: term vs sum, counting n, and spotting the sequence
Proof: testing a few values instead of proving the general case
Confusing common difference d with common ratio r
Using the arithmetic sum formula for a geometric series

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Recognise an arithmetic sequence from its common difference and use uₙ = u₁ + (n − 1)d to find a term, the first term or the number of terms.
Use Sₙ = u₁(rⁿ − 1)/(r − 1) to find the sum of a finite geometric series and solve problems for an unknown ratio or number of terms.
Solve exponential equations by writing both sides with the same base, such as 4ˣ⁺¹ = 8ˣ, and check answers by substitution.
Calculate binomial coefficients ⁿCᵣ with and without technology and use them to expand (a + b)ⁿ for larger n, writing out the first few terms.
Use the general term ⁿCᵣ aⁿ⁻ʳ bʳ to find a specified term or coefficient of (a + b)ⁿ without writing out the whole expansion.
Write clear LHS-to-RHS proofs of results such as the sum of the first n odd numbers or algebraic identities, using correct notation.

The printable sheet

Number and algebra knowledge organiser for IB Maths AA SL: one A4 page of key definitions, formulas, a worked example and common mistakes — Number and algebra knowledge organiser (IB Maths AA SL), A4. Download the PDF.

Revise it next

Other IB Maths AA SL topics: Functions · Geometry and trigonometry · Statistics and probability · Calculus · All IB Maths AA SL organisers