IB Mathematics: Analysis and Approaches
Number and Algebra Formulae for AA SL and HL
A complete, exam-focused guide to the shared Topic 1 content studied by both AA SL and AA HL students: sequences, series, finance, exponents, logarithms, proof, infinite geometric sums and the binomial theorem.
Reviewed for the current IB Mathematics: Analysis and Approaches course (first assessment 2021). A separate note near the end explains the revised course first assessed in May 2029.
This is the common Number and Algebra foundation for both Mathematics: Analysis and Approaches standard level and higher level. An AA HL student does not skip this material; HL adds further content after it. That distinction is important when you organize revision. This page covers the current SL 1.1–1.9 sequence, while complex numbers, De Moivre’s theorem, advanced proof and systems of equations belong in the companion AA HL-only Numbers and Algebra guide.
A formula sheet is most useful when it is connected to meaning. You should be able to identify each variable, state the conditions, choose an efficient representation and check whether the result makes sense. IB questions regularly combine ideas: a geometric sequence may model compound growth, a logarithm may solve for time, or a binomial coefficient may appear inside probability. The goal is therefore method selection, not memorizing disconnected lines.
Current AA SL/HL Number and Algebra syllabus map
The syllabus references make it easier to match class notes, question-bank filters and revision tasks. They do not mean that examination questions stay inside one numbered box. Topic 1 connects naturally to functions, statistics, calculus and modelling. For instance, arithmetic sequences correspond to discrete linear change, geometric sequences correspond to discrete exponential change, and logarithms invert exponential relationships.
| Current item | Main content | Typical exam action |
|---|---|---|
| SL 1.1 | Operations with numbers in the form a×10k | Calculate and report a valid scientific-notation answer |
| SL 1.2 | Arithmetic sequences and series, sigma notation and applications | Find a term, common difference, number of terms or finite sum |
| SL 1.3 | Geometric sequences and series, sigma notation and applications | Find a ratio, term or finite sum and interpret a growth/decay model |
| SL 1.4 | Compound interest, annual depreciation and real value | Model repeated percentage change and solve a financial problem |
| SL 1.5 | Integer exponent laws; introduction to base-10 and natural logarithms | Simplify powers and evaluate logarithms with technology |
| SL 1.6 | Simple numerical/algebraic deductive proof; equality and identity | Transform one side logically and check a result |
| SL 1.7 | Rational exponents, logarithm laws, change of base and exponential equations | Rewrite, simplify or solve exactly/numerically |
| SL 1.8 | Infinite convergent geometric series | Check |r|<1 and calculate or build a sum to infinity |
| SL 1.9 | Binomial theorem for n∈ℕ; Pascal’s triangle and combinations | Expand, find a term/coefficient or solve for an index |
These sections assume that you are comfortable with algebraic substitution, fractions, rearranging equations and calculator settings. If that foundation needs work, begin with the AA SL/HL prior-learning formulae before attempting timed mixed questions.
SL 1.1: scientific notation
Scientific notation writes a non-zero number as a×10k, where 1≤|a|<10 and k is an integer. It makes very large and very small values easier to compare and calculate. The coefficient must be normalized: 42×105 is mathematically equal to 4.2×106, but only the second expression is standard scientific notation.
When multiplying, multiply the coefficients and add the exponents. When dividing, divide the coefficients and subtract the exponents. Addition and subtraction require the same power of ten first. Normalize only after performing the operation.
Worked example: mixed scientific-notation calculation
Evaluate (6.4×107)(3×10−4)/(8×102).
The exponent in 3×10−4 is −4, so adding it reduces the power. A common error is to read “7−4−2” without first recognizing that multiplication adds the signed exponent.
In formal written answers, use ×10k rather than calculator E-notation. A screen may display 5.2E30, but the official guide explicitly expects 5.2×1030. Retain enough exact information during working and round only at the requested stage.
SL 1.2: arithmetic sequences and series
An arithmetic sequence changes by a constant difference d. If the first term is u1, the nth term is found by adding d exactly n−1 times. A series is the sum of sequence terms. Keep “term” and “sum” separate: un is one value, while Sn is the total of the first n values.
Sn=n/2[2u1+(n−1)d]=n/2(u1+un)
Use the first sum form when you know u1, d and n. Use the second when you know the first and last terms. Sigma notation compresses a sum: Σ from r=1 to n of [u1+(r−1)d]. The running variable is a label; changing r to k does not change the sum if the limits and expression are adjusted consistently.
Worked example: find a term and a total
A theatre has 18 seats in the first row and three more seats in each successive row. There are 24 rows. Find the seats in the last row and the total capacity.
S24=24/2(18+87)=12(105)=1260.
The model is arithmetic because the absolute increase is constant. State the interpretation: the final row has 87 seats and the theatre has 1,260 seats in these rows.
Solving backwards
Some questions provide a later term or sum and ask for n, d or the first term. Substitute known values before expanding. Because n counts terms, it must normally be a positive integer. If algebra gives n=10.6 in a contextual question, decide whether the threshold is first reached at n=11 rather than reporting a fractional term number.
Arithmetic models are useful when change is approximately constant. The official guide also expects analysis and prediction where real data are not perfectly arithmetic, so you may need to estimate a common difference and discuss limitations. A model can be useful without matching every observation exactly.
SL 1.3: geometric sequences and series
A geometric sequence changes by a constant ratio r. Each term is the previous term multiplied by r. If 0<r<1, positive terms decay toward zero; if r>1, they grow; a negative r alternates signs. A geometric model represents constant proportional change, which is why it connects to population, disease spread, salary change, radioactive decay and finance.
Sn=u1(1−rn)/(1−r), r≠1
An equivalent finite-sum form is u1(rn−1)/(r−1). Choose one and keep its numerator and denominator signs paired. If r=1, every term equals u1 and Sn=nu1, so the standard fraction is unnecessary.
Worked example: geometric decay
A machine is worth $48,000 and retains 82% of its value each year. Find its value after six yearly reductions.
The phrase “after six reductions” gives exponent 6. If the initial value is called year 0, the sequence term at year 6 corresponds to seven listed values but six multiplications. Draw a short timeline whenever the indexing is ambiguous.
Recognizing arithmetic versus geometric structure
| Question | Arithmetic | Geometric |
|---|---|---|
| What stays constant? | Difference un+1−un | Ratio un+1/un |
| Typical language | adds, increases by an amount | multiplies, grows by a percentage |
| Graph of term against n | Discrete points on a line | Discrete exponential pattern |
| nth term | u1+(n−1)d | u1rn−1 |
SL 1.4: compound interest, depreciation and real value
Financial applications use geometric change because interest or depreciation is applied to the current balance, not repeatedly to the original amount. A nominal annual rate of R% compounded m times per year gives a periodic multiplier 1+R/(100m). For depreciation, replace the plus sign with a minus sign when R is the annual reducing rate.
With annual rate R% compounded m times yearly: A=P[1+R/(100m)]mt.
P is principal, i is the rate per compounding period, n is the number of periods and A is the accumulated amount. Units must match. A monthly rate requires a number of months; an annual rate divided by 12 is used with 12t monthly periods when the question describes nominal annual compounding in that way.
Worked example: monthly compounding
$7,500 is invested at a nominal annual rate of 4.8%, compounded monthly, for five years. Find the balance.
Do not use 0.048 as the monthly rate or use exponent 5 with a monthly multiplier. Store full calculator precision and round the final currency amount as instructed.
Real value and inflation
Money can grow in nominal terms while losing purchasing power. If inflation is j per period, divide the future nominal amount by (1+j)n to express it in today’s purchasing power. Equivalently, compare the investment multiplier with the inflation multiplier.
A question may require a GDC finance application, spreadsheet or solver. Technology is permitted where specified, but define inputs and interpret outputs. A negative payment in a finance solver often represents money leaving an account; sign conventions matter.
SL 1.5: exponent laws and introductory logarithms
Exponent laws describe repeated multiplication and its extensions. They work for non-zero bases where division or negative exponents occur. Parentheses control the base: −24=−16, while (−2)4=16.
am/an=am−n, a≠0
(am)n=amn
(ab)n=anbn
a0=1, a≠0 and a−n=1/an.
A logarithm answers an exponent question. The statement logab=x means exactly ax=b, where a>0, a≠1 and b>0. At this introductory stage, base 10 and base e are central. Log without a written base commonly denotes base 10 in calculator contexts; ln denotes loge.
ln x=logex.
Worked example: interpret before calculating
Evaluate log100.001 and ln(e5).
Because 10−3=0.001, log100.001=−3. Because ln and the exponential with base e are inverse operations, ln(e5)=5. These exact conclusions are preferable to decimal calculator output.
SL 1.6: simple deductive proof, equality and identity
A deductive proof uses accepted facts and valid algebraic steps to establish a conclusion. In a left-hand-side-to-right-hand-side identity proof, begin with one side only and transform it until it becomes the other. Do not write the target equality at the start and manipulate both sides as if it were already known.
An equation is true only for particular solution values. An identity is true for every value in its domain and is often written with ≡. For example, (x−3)2+5≡x2−6x+14 because expansion verifies it for all real x. By contrast, x+3=8 is an equation with solution x=5.
Worked example: an algebraic identity
Show that (x+1)/(x−1)−2/(x−1)≡1 for x≠1.
The restriction x≠1 must remain because the original expressions are undefined there. Cancellation does not restore an excluded domain value.
Checking is not proving
Substituting several numerical values can find an error or support a conjecture, but it does not prove a universal identity. Use algebra for the proof, then substitute a sensible value as a check. Similarly, when solving an equation, substitute the candidate into the original equation, especially after squaring, clearing denominators or using logarithms.
SL 1.7: rational exponents and logarithm laws
Rational exponents connect powers and roots. For appropriate real values, a1/n is the nth root of a and am/n=(a1/n)m. When n is even and the real principal root is intended, the root is non-negative. Domain restrictions become important when negative bases or fractional powers appear.
Logarithm laws follow from exponent laws. They apply only when every logarithm argument is positive.
loga(x/y)=logax−logay
loga(xm)=m logax
logax=logbx/logba.
The first law does not say log(x+y)=log x+log y. There is no corresponding law for a sum inside a logarithm. The power law moves an exponent in front; it does not multiply the argument.
Worked example: solve an exponential equation
Solve 32x−1=17.
Take logarithms of both sides:
x=[1+ln17/ln3]/2≈1.789.
Using log base 10 gives the same answer because change of base preserves the ratio. Substitute the unrounded value to check.
Logarithms in sequence and finance questions
When an unknown appears in an exponent, logarithms usually isolate it. If P(1+i)n=A, then n=ln(A/P)/ln(1+i). In a discrete-time context, the calculated n may need to be rounded up to the first whole period at which a threshold is exceeded. State that interpretation rather than reporting only a decimal.
SL 1.8: infinite convergent geometric series
A finite geometric sum has a last term. An infinite geometric series continues without end, yet it can approach a finite limit when successive terms shrink sufficiently. The required condition is |r|<1. If r=1, terms never shrink; if r≤−1 or r≥1, the partial sums do not settle to a finite value.
The condition is part of the formula. Always identify the first term and common ratio from the actual series, then test convergence before substituting.
Worked example: recurring decimal
Express 0.272727… as a fraction using a geometric series.
Write 0.272727…=0.27+0.0027+0.000027+… . The first term is 27/100 and r=1/100.
Because |1/100|<1, the series converges. The fraction is exact.
Worked example: bouncing distance
A ball is dropped 5 m and rebounds to 60% of each previous height. Find the total vertical distance travelled.
The initial 5 m drop occurs once. Every rebound height is travelled upward and downward. The rebound heights form 3+1.8+1.08+… with r=0.6.
Do not double the initial drop. A diagram makes the one-way and two-way parts clear.
SL 1.9: binomial theorem for positive integer powers
The binomial theorem expands (a+b)n for a positive integer n. The coefficients can be read from Pascal’s triangle or calculated using combinations. Unlike the fractional and negative expansions studied in the HL-only extension, this shared-course expansion is finite and ends after n+1 terms.
Tr+1=C(n,r)an−rbr.
Be careful with term numbering. The r=0 term is the first term, so the general term indexed by r is Tr+1. In each successive term, the power of a decreases by one and the power of b increases by one; their exponents always add to n.
Worked example: full expansion
Expand (2x−3)4.
=16x4−96x3+216x2−216x+81.
Keep −3 inside parentheses when raising it to a power. Alternating signs follow from the negative second term.
Worked example: one coefficient only
Find the coefficient of x3 in (1−2x)7.
The x3 term occurs when r=3:
Therefore the coefficient is −280. There is no need to expand all eight terms.
Pascal’s triangle is efficient for small powers, while C(n,r) scales better and is supported by technology. You should be able to use both. The theorem also connects to probability, where the same coefficients count ways to arrange successes and failures.
Formula booklet: what is provided and what you must do
The official guide says that all formulae required for the course appear in the mathematics formula booklet and that students receive a clean copy during examinations. “Provided” does not mean “automatic.” The booklet cannot diagnose the model, align time units, choose the right term index, check a convergence condition or explain a proof.
| Area | Booklet support | Your responsibility |
|---|---|---|
| Arithmetic sequences | nth-term and finite-sum relationships | Identify u1, d and the correct term count |
| Geometric sequences | nth-term, finite-sum and infinite-sum relationships | Identify r, distinguish finite/infinite and check |r|<1 |
| Finance | Compound-growth structure supports modelling | Convert rates and periods consistently and interpret cash flow |
| Exponents/logarithms | Standard identities and relationships | Respect domains, select a base and solve the requested equation |
| Proof | No formula writes the logical argument | Begin from one side, justify transformations and retain restrictions |
| Binomial theorem | General expansion and combinations notation | Track r, signs, coefficients and the requested power |
Practise with the exact booklet edition supplied by your school. Locate the correct section, identify the variables and state the restrictions before calculating. Do not annotate the examination booklet; the official requirement is a clean copy.
How Number and Algebra appears in AA examinations
AA questions reward algebraic reasoning and exact communication. On a no-technology paper, you may need to simplify, prove or derive a result by hand. On a technology paper, a GDC can generate sequence values, solve an exponential equation or use a financial application, but you should still define the model and report the conclusion clearly.
Command terms matter. “Write down” may require little working; “find” requires a result supported by a valid method; “show that” requires a convincing chain that reaches a supplied result without circular reasoning; “hence” asks you to use the previous result efficiently. Preserve exact values such as logarithms or fractions unless a decimal approximation is requested.
A six-step method for structured questions
- Decode: identify the command term, target quantity and required form.
- Classify: decide whether the structure is arithmetic, geometric, exponential, logarithmic, proof-based or binomial.
- Define: write variables with units and align the time/index convention.
- Calculate: show a formula or mathematical route and keep guard digits.
- Check: substitute, estimate, test the domain or compare with the unrestricted model.
- Conclude: answer in context with correct rounding and notation.
Frequent errors and corrections
Wrong sequence type
Test consecutive differences and ratios. A constant percentage change is geometric, even if the question says “increases each year.”
Off-by-one exponent
un uses n−1 multiplications from the first term. A financial timeline may begin at time zero; sketch it.
Rate-period mismatch
Monthly compounding needs a monthly rate and a number of months. Convert both parts together.
Invalid logarithm step
There is no sum law for logarithms. Check that every log argument is positive.
Infinite-sum condition omitted
Write |r|<1 before using S∞. If it fails, the series does not have that finite sum.
Binomial sign lost
Keep the complete negative term in parentheses before applying its power.
A productive error log records the reason for a lost mark: concept choice, algebra, calculator entry, notation, rounding or interpretation. Redo the question from a blank page a day later. Reading a corrected solution is less effective than reconstructing it.
Exact answers, decimal answers and sensible checking
Keep an answer exact while the mathematics is exact. Fractions, powers, logarithms and surds preserve information that an early decimal can lose. If the final instruction asks for three significant figures, store the unrounded calculator result and round once at the end. For money, follow the requested currency precision; for a number of whole periods, decide whether the context requires the next integer rather than ordinary rounding. A threshold problem asking when a balance first exceeds a value usually requires rounding the calculated time up and verifying the periods on both sides.
Estimation is a fast defence against calculator errors. A 5% annual increase for a few years should not multiply an amount by ten; a decaying geometric sequence with ratio 0.7 should not grow; a sum to infinity with positive terms must exceed its first term; and a binomial coefficient is an integer. Before moving on, ask whether the sign, scale, units and direction of change agree with the problem. These checks take seconds and often recover marks that would otherwise be lost to a mistyped exponent or rate.
Fourteen-day revision plan
| Days | Focus | Mastery evidence |
|---|---|---|
| 1 | Scientific notation and exponent laws | Operations are normalized and written without E-notation |
| 2–3 | Arithmetic sequences, series and sigma notation | You can solve forward and backward and explain indexing |
| 4–5 | Geometric sequences and finite sums | You can identify r and distinguish term from sum |
| 6–7 | Finance, depreciation and inflation | Rates, periods and interpretations are consistent |
| 8 | Simple proof, equality and identity | You can transform one side and retain domain restrictions |
| 9–10 | Rational exponents, logarithm laws and equations | You can solve unknown-exponent problems and check domains |
| 11 | Infinite geometric series | You test convergence before calculating the sum |
| 12–13 | Binomial expansions and individual coefficients | You control r, signs, powers and term numbering |
| 14 | Timed mixed paper and error repair | You select a method without a topic label |
At the end of each block, attempt one unfamiliar problem without notes. Use official specimen or past-paper material available through your school, mark against the correct session’s markscheme and rewrite only the steps that lost marks. For the function connections, continue with Functions Formulae AA SL and HL.
Current course versus first assessment 2029
The current AA course was first assessed in May 2021. The IB has announced an updated course for first teaching in August 2027 and first assessment in May 2029. Its published update says selected material is removed rather than adding new content. For AA standard and higher level, financial applications are among the announced removals; the assessment mark totals also change.
This guide intentionally follows the current SL 1.1–1.9 sequence. If your final examination is May 2029 or later, check the new subject guide and formula booklet supplied by your school. Do not combine a current-course numbered topic list with a 2029 assessment table. Even where a topic is no longer directly assessed, its mathematics may remain useful for modelling and other subjects.
Frequently asked questions
Is this content required for both AA SL and AA HL?
Yes. SL 1.1–1.9 is the shared Number and Algebra foundation. AA HL students study this material and then add the higher-level-only extension.
What does the AA HL-only extension add?
The current extension adds counting principles, generalized binomial expansions, partial fractions, complex numbers, De Moivre’s theorem, advanced proof methods and systems of linear equations. Use the linked HL-only guide for those topics.
Are all formulas provided in the IB Mathematics formula booklet?
The official guide says all required formulae are in the mathematics formula booklet and a clean copy is available in examinations. You still need to choose the correct relationship, understand its variables and conditions, and show the requested method.
How can I tell arithmetic and geometric sequences apart?
Arithmetic sequences have a constant difference. Geometric sequences have a constant non-zero ratio. Constant absolute growth suggests arithmetic; constant percentage growth suggests geometric.
When can I use the sum-to-infinity formula?
Only for a geometric series with |r|<1. Identify the ratio and state the condition before using S∞=u1/(1−r).
Why does the arithmetic nth term contain n−1?
The first term needs zero additions of d. Moving from term 1 to term n involves n−1 equal steps.
Why are logarithms useful?
They invert exponentials. When an unknown appears in an exponent, taking logarithms often brings it down as a multiplier so the equation can be solved.
Can I use my calculator’s finance or sequence applications?
Use technology on papers and questions where it is permitted. Define inputs, observe sign conventions and interpret the result. A screen value alone may not satisfy a command asking for reasoning.
Is compound interest still in the syllabus?
It is in the current course described here. The IB’s update says financial applications are removed from the revised course first assessed in May 2029, so follow the guide for your examination year.
How many terms are in a binomial expansion?
For (a+b)n with positive integer n, there are n+1 terms, indexed by r=0,1,…,n.
Should I memorize formulas?
Build familiarity, but prioritize recognition and use. Practise locating a formula in the official booklet, naming each variable, stating restrictions and applying it to unfamiliar questions.
What should I revise next?
Review the prior-learning guide if algebra is weak, move to the functions guide for exponential/logarithmic graphs, and use the HL-only guide if you take higher level.
Official references and related RevisionTown guides
This guide was checked against the official IB Mathematics: Analysis and Approaches guide, the IB’s Diploma Programme mathematics overview, and the official 2029 curriculum update. RevisionTown is an independent educational resource and is not affiliated with or endorsed by the International Baccalaureate Organization.
Continue with Prior Learning Formulae AA SL and HL, Functions Formulae AA SL and HL, or Numbers and Algebra Formulae AA HL Only.
