Number and Algebra

Q: What is the difference between an arithmetic and a geometric sequence?

An arithmetic sequence adds a constant difference $d$ between terms, while a geometric sequence multiplies by a constant ratio $r$. Check consecutive differences for arithmetic and consecutive ratios for geometric.

Q: Is the binomial theorem examinable at SL in Maths AI?

No — the binomial theorem is HL-only content in Maths AI. SL students are not assessed on it.

Q: How do I solve an exponential equation where the variable is in the exponent?

Isolate the exponential term completely on one side of the equation, then take $\ln$ or $\log$ of both sides to bring the variable down.

Q: When does a geometric series have a sum to infinity?

Only when $|r|<1$. If the common ratio's absolute value is 1 or greater, the terms don't shrink and no finite sum exists.

Q: Why can't I split $\log(a+b)$ into $\log a + \log b$?

Log laws only apply to products, quotients, and powers — there is no log law for addition or subtraction inside the bracket.

Q: What does $T_{r+1}$ mean in the binomial theorem?

It's the general term formula, and the term number is always one more than $r$ — so $r=3$ gives the 4th term, not the 3rd.

Exponentials, logs, binomial expansions, and sequences — the backbone of IB Maths AI, decoded fast.

Subject

Maths AI

Curriculum

IB Diploma Programme

Grade

Topic

Number and Algebra

Reading

9 min

Difficulty

Advanced

Quick facts

Difficulty

★★★★☆

Exam weight

Largest strand — recurs in Paper 1 & Paper 2 almost every sitting

Prerequisites

Index laws, basic algebraic manipulation

You'll learn

Exponential/log models, binomial expansions, arithmetic & geometric series

Revision time

45–60 min

IB Maths AI Number and Algebra is where growth, decay, and pattern-spotting collide with algebra — it's the strand that turns real contexts like population growth, drug concentration, and theatre seating into solvable equations. Whether you're isolating an exponential term to take logs, expanding a binomial with the general term formula, or spotting whether a sequence adds or multiplies, the same three-step approach works every time: identify the pattern, choose the matching formula, then solve. This topic underpins financial maths and modelling questions elsewhere on the paper, so getting comfortable with exponential and logarithmic functions, arithmetic and geometric sequences, and (for HL) the binomial theorem pays off across the whole exam. This teaser covers the five ideas examiners return to most, with the traps that cost real marks — the full revision notes go deeper into every formula and worked example.

What you’ll be able to do

✓Apply exponent and logarithm laws to solve equations

✓Reconstruct an exponential model from two data points

✓Expand $(a+b)^n$ using the binomial theorem and general term

✓Identify and use the nth term and sum formulas for arithmetic sequences

✓Identify and use the nth term and sum formulas for geometric sequences

✓Determine when a geometric series has a sum to infinity

✓Translate real-world contexts into sequence or exponential models

✓Avoid the most common algebraic and indexing mistakes in this topic

Exponential and Logarithmic Functions

An exponential function like $y = a b^{x}$ or $y = A e^{k x}$ changes by a constant factor every time $x$ increases by 1 — that's the entire idea, everything else is manipulation. Logarithms exist purely to pull a trapped variable down out of the exponent, so whenever $t$ sits in a power, isolate the exponential term first and then take $ln$ or $lo g$ of both sides. Two data points on a model are just simultaneous equations wearing an exponential disguise.

Exponential growth curve alongside a logarithm equation showing the variable isolated before taking log

Operation	Exponent form	Matching log law
Multiplication	$a^{m} \times a^{n} = a^{m + n}$	$lo g_{b} (x y) = lo g_{b} x + lo g_{b} y$
Division	$a^{m} / a^{n} = a^{m - n}$	$lo g_{b} (x / y) = lo g_{b} x - lo g_{b} y$
Power	$(a^{m})^{n} = a^{mn}$	$lo g_{b} (x^{n}) = n lo g_{b} x$

Exam tip

'Exact value' means leave your answer as a log or fraction — a rounded decimal loses the final mark.

Common mistake

Writing $lo g (a + b) = lo g a + lo g b$ : there is no log law for a sum inside the bracket, only for products, quotients, and powers.

Mini summary

Isolate the exponential, then take logs — never log across a sum.

Binomial Theorem (HL only)

The binomial theorem expands $(a + b)^{n}$ without multiplying it out term by term — essential once $n \geq 4$ , where hand expansion becomes error-prone. Pascal's triangle gives coefficients by eye for small $n$ , while $(r n)$ (your GDC's nCr button) works for any $n$ instantly. This is a reliable stand-alone 4–6 mark question at HL — SL students never see it.

$Pascal's triangle next to the binomial theorem formula with the general term T_{r+1} highlighted$

Exam tip

Set the power of $b$ equal to the target power of $x$ to find $r$ , then always label the term $T_{r + 1}$ , never $T_{r}$ .

Common mistake

Forgetting to raise the whole bracketed term to the power $r$ — e.g. writing $(- 3 x)^{3}$ as $- 3 x^{3}$ instead of correctly cubing both the coefficient and the variable.

Mini summary

Powers of $a$ and $b$ always sum to $n$ ; the term with $b^{r}$ is $T_{r + 1}$ .

Arithmetic Sequences and Series

An arithmetic sequence adds a fixed common difference $d$ to get from one term to the next — check that $u_{2} - u_{1} = u_{3} - u_{2}$ and so on. Every formula needs just two ingredients, $u_{1}$ and $d$ , and any term or partial sum falls straight out of those. IB loves hiding these in context — theatre rows, stacked cans, fixed pay rises — so translate the words into $u_{1}$ and $d$ before touching a formula.

Number line showing an arithmetic sequence with equal jumps of d between each term

Exam tip

Self-check your nth-term formula with $n = 1$ : it must return $u_{1}$ exactly, catching the classic $(n - 1)$ vs $n$ slip instantly.

Common mistake

Writing $u_{n} = u_{1} + n d$ instead of $u_{1} + (n - 1) d$ — this silently shifts every term forward by one position.

Mini summary

Arithmetic = constant addition of $d$ ; always $(n - 1)$ in the nth-term formula.

Geometric Sequences and Series

A geometric sequence multiplies by a fixed ratio $r$ each step — check $u_{2} / u_{1} = u_{3} / u_{2}$ and so on. It's the discrete cousin of the exponential function: $u_{n} = u_{1} r^{n - 1}$ is exactly $a \cdot b^{x}$ with integer $x$ . Any 'changes by a fixed percentage' context — car depreciation, compound interest, colony growth — is a geometric sequence with $r = 1 \pm \frac{percentage}{100}$ .

Diagram comparing arithmetic addition versus geometric multiplication between sequence terms

Exam tip

Spot percentage-change wording immediately: convert it to a ratio $r$ before writing any formula.

Common mistake

Confusing a geometric context for an arithmetic one when a question mentions 'increases by 5%' rather than 'increases by 5' — the first is multiplicative (geometric), the second additive (arithmetic).

Mini summary

Geometric = constant multiplication by $r$ ; links directly to exponential models.

Sum to Infinity & Real-World Applications

A geometric series only has a finite sum to infinity when $∣ r ∣ < 1$ — outside that range the terms never shrink and the total never settles. This condition is exactly why compound interest and depreciation problems (both geometric) behave so differently from a shrinking geometric sum like a bouncing ball's total distance. Recognising whether a context is exponential, arithmetic, or geometric — and whether $∣ r ∣ < 1$ applies — is the final skill examiners test across every real-world Number and Algebra question.

Geometric series terms shrinking toward zero with the condition |r| less than 1 highlighted

Exam tip

Before summing to infinity, always state and check $∣ r ∣ < 1$ explicitly — this is often a required method line, not just a formality.

Common mistake

Applying a sum-to-infinity formula to a series where $∣ r ∣ \geq 1$ , which produces a meaningless or undefined result.

Mini summary

Sum to infinity exists only for $∣ r ∣ < 1$ ; always verify the condition before using the formula.

Quick formula sheet

f (x) = a \cdot b^{x} or A e^{k x}

Exponential function — output multiplies by a constant factor per unit increase in x — Same shape, new costume — every model with constant % change is this in disguise

lo g_{b} (x y) = lo g_{b} x + lo g_{b} y, lo g_{b} (\frac{x}{y}) = lo g_{b} x - lo g_{b} y, lo g_{b} (x^{n}) = n lo g_{b} x

Laws of logarithms — only for products, quotients, powers — No law for sums inside the bracket — ever

lo g_{b} a = \frac{ln a}{ln b}

Change of base — needed when your calculator only has log/ln buttons

(a + b)^{n} = \sum_{r = 0}^{n} (r n) a^{n - r} b^{r}

Binomial theorem for positive integer n (HL only) — Powers of a and b always sum to n

T_{r + 1} = (r n) a^{n - r} b^{r}

General term of a binomial expansion — term number is always r+1

u_{n} = u_{1} + (n - 1) d

nth term of an arithmetic sequence — Test n=1: must give u1

S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) d) = \frac{n}{2} (u_{1} + u_{n})

Sum of first n terms of an arithmetic series

u_{n} = u_{1} r^{n - 1}

nth term of a geometric sequence — the discrete cousin of

a \cdot b^{x}

S_{n} = \frac{u _{1} ( 1 - r ^{n} )}{1 - r}, r \neq = 1

Sum of first n terms of a geometric series

Practice questions

Easy

Simplify $\frac{a ^{5} \times a ^{- 2}}{a ^{3}}$ using the laws of exponents.
Find the 10th term of the arithmetic sequence with $u_{1} = 4$ and $d = 3$ .
State the ratio $r$ of the geometric sequence $5, 15, 45, 135, \dots$

Medium

Solve $2 e^{0.5 t} = 18$ for $t$ , giving your answer to 3 significant figures.
Find the sum of the first 15 terms of an arithmetic series with $u_{1} = 7$ and $d = - 2$ .
Find the coefficient of $x^{2}$ in the expansion of $(1 + 2 x)^{6}$ .

Challenge

A population is modelled by $P (t) = P_{0} e^{k t}$ . Given $P (0) = 40000$ and $P (15) = 52000$ , find the exact value of $k$ in the form $\frac{1}{15} ln (a)$ .
A geometric series has $u_{1} = 8$ and sum to infinity $S_{\infty} = 20$ . Find $r$ and hence $u_{4}$ .
For $(3 - x)^{7}$ , determine which term contains $x^{4}$ and find its coefficient.

Frequently asked questions

What is the difference between an arithmetic and a geometric sequence?+

An arithmetic sequence adds a constant difference $d$ between terms, while a geometric sequence multiplies by a constant ratio $r$ . Check consecutive differences for arithmetic and consecutive ratios for geometric.

Is the binomial theorem examinable at SL in Maths AI?+

No — the binomial theorem is HL-only content in Maths AI. SL students are not assessed on it.

How do I solve an exponential equation where the variable is in the exponent?+

Isolate the exponential term completely on one side of the equation, then take $ln$ or $lo g$ of both sides to bring the variable down.

When does a geometric series have a sum to infinity?+

Only when $∣ r ∣ < 1$ . If the common ratio's absolute value is 1 or greater, the terms don't shrink and no finite sum exists.

Why can't I split $\log(a+b)$ into $\log a + \log b$?+

Log laws only apply to products, quotients, and powers — there is no log law for addition or subtraction inside the bracket.

What does $T_{r+1}$ mean in the binomial theorem?+

It's the general term formula, and the term number is always one more than $r$ — so $r = 3$ gives the 4th term, not the 3rd.

Get the full IB Maths AI Number and Algebra notes

Complete worked examples for every exponential, log, sequence, and binomial question type Step-by-step traps and fixes for the exact mistakes examiners penalise most Full formula derivations plus mock exam-style questions with guided solutions Built specifically for IB DP Maths AI at the level Paper 1 and Paper 2 actually test

Get the Number and Algebra notes on RevisionPrep →

Maths AI: Geometry and Trigonometry (HL)Maths AI: Statistics and Probability — Regression & Hypothesis Testing Maths AI: Statistics and Probability — Correlation & Chi-Squared Maths AI: Geometry and Trigonometry (SL)