Syllabus map

AA SL Number and Algebra syllabus

The checklist below names the Standard Level syllabus points for Topic 1. Higher Level extensions are covered in the separate HL guide.

  1. 1.1Operations in scientific notation / standard form $a \times 10^k$, SI metric prefixes
  2. 1.2Arithmetic sequences and series, common difference, general term $u_n$, sum $S_n$
  3. 1.3Geometric sequences and finite series, common ratio $r$, term $u_n$, sum $S_n$
  4. 1.4Financial applications: compound interest, nominal vs effective rate, depreciation
  5. 1.5Integer exponents and introduction to logarithms
  6. 1.6Simple deductive proof, algebraic identities, disproof by counterexample
  7. 1.7Rational exponents, laws of logarithms, change of base, exponential equations
  8. 1.8Sum of infinite convergent geometric sequences $S_\infty = \frac{u_1}{1 - r}$ with $|r| < 1$
  9. 1.9Binomial theorem for $n \in \mathbb{N}$, Pascal's triangle, general term $\binom{n}{r} a^{n-r} b^r$

Study by topic

Topic 1 Interactive Study Guides

All 6 lesson guides are equipped with our gold-standard 4-tab interface: Study Notes (pointwise concepts, SVGs & worked examples), Practice Exercises (10 tiered problems with collapsible solutions), Formula Sheet (formula booklet & pitfall alerts), and Mini Test (interactive MCQs & exam FRQs).

1.1 Standard Form Scientific notation $a \times 10^k$, SI metric prefixes, rounding rules, estimation, and error bounds. Includes 4-tab interactive notes; includes practice questions and a mini test. Open Interactive Guide 1.2 Arithmetic Sequences & Series Common differences, $n$th term formulas, partial sums, and linear mathematical modeling. Includes interactive 4-tab guide, 10 tiered problems, and mini test. Open Interactive Guide 1.3, 1.8 Geometric Sequences & Infinite Sums Common ratios, finite sums, convergence criterion $|r| < 1$, and infinite geometric series limits. Complete with interactive 4 tabs, diagrams, and exam questions. Open Interactive Guide 1.4 Financial Applications of Sequences Compound interest, compounding frequencies, nominal vs effective annual rates, depreciation, and real-world geometric growth with 10 tiered exercises. Open Interactive Guide 1.5, 1.7 Exponents and Logarithms Laws of exponents, rational radicals, logarithmic identities, change of base, and solving exponential equations with 4-tab interactive study guide. Open Interactive Guide 1.6, 1.9 Simple Proof & Binomial Theorem Direct deductive algebraic proofs, disproof by counterexample, Pascal's triangle, and binomial expansions $(a+b)^n$ with 10 tiered exercises and diagnostic quiz. Open Interactive Guide

Learn the ideas

Core Mathematical Concepts

Standard form and sensible precision

  • Write a non-zero number as $a \times 10^k$, where $1 \le |a| < 10$ and $k \in \mathbb{Z}$. Always normalize the leading coefficient $a$ after multiplication or division.
  • In IB examinations, give non-exact numerical answers to 3 significant figures unless specified otherwise. Keep intermediate values unrounded in memory.

Arithmetic and geometric sequences

  • An arithmetic sequence has a constant common difference $d$: $u_n = u_1 + (n - 1)d$. Its sum is $S_n = \frac{n}{2}[2u_1 + (n - 1)d] = \frac{n}{2}(u_1 + u_n)$.
  • A geometric sequence has a constant common ratio $r$: $u_n = u_1 r^{n-1}$. For $r \ne 1$, $S_n = \frac{u_1(1 - r^n)}{1 - r} = \frac{u_1(r^n - 1)}{r - 1}$.
  • An infinite geometric series converges to a finite sum if and only if $|r| < 1$. Its sum to infinity is $S_\infty = \frac{u_1}{1 - r}$.
  • Compound interest over $k$ periods per year at nominal annual rate $r\%$ is modeled geometrically: $FV = PV \left(1 + \frac{r}{100k}\right)^{kn}$.

Exponents and logarithms

  • Index laws: $a^x a^y = a^{x+y}$, $\frac{a^x}{a^y} = a^{x-y}$, $(a^x)^y = a^{xy}$, $a^{-x} = \frac{1}{a^x}$, $a^{m/n} = \sqrt[n]{a^m}$.
  • Logarithmic definition: $\log_a x = y \iff a^y = x$ with base $a > 0, a \ne 1$ and argument $x > 0$.
  • Log laws: $\log_a(xy) = \log_a x + \log_a y$, $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$, $\log_a(x^k) = k \log_a x$, and change of base $\log_a b = \frac{\ln b}{\ln a}$.

Simple proof and the binomial theorem

  • Direct proof requires rigorous algebraic deductive reasoning starting from one side (LHS) to arrive at the other (RHS). Examples do not constitute proof.
  • A universal conjecture is disproved by providing a single counterexample within the specified domain.
  • The binomial expansion for $n \in \mathbb{N}$ is $(a + b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} b^r$, where the general term is $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ with $\binom{n}{r} = \frac{n!}{r!(n-r)!}$.

Follow the reasoning

Worked examples

These original examples show common reasoning steps across Topic 1 concepts.

Worked example 1

Find a term and a partial sum

Question. An arithmetic sequence begins $7, 11, 15, \dots$ Find $u_{12}$ and the sum of the first 12 terms.

  1. The first term is $u_1 = 7$ and the common difference is $d = 11 - 7 = 4$.
  2. $u_{12} = 7 + (12 - 1)(4) = 7 + 44 = 51$.
  3. $S_{12} = \frac{12}{2}[2(7) + 11(4)] = 6(14 + 44) = 6(58) = 348$.

Answer: $u_{12} = 51$ and $S_{12} = 348$.

Worked example 2

Model compound growth

Question. \$5,000 is invested at 4% annual interest compounded quarterly. Find its future value after 3 years, to the nearest dollar.

  1. $PV = 5000$, $r = 4\%$, compounding periods per year $k = 4$, and $n = 3$ years. Total compounding periods $N = 4 \times 3 = 12$.
  2. Quarterly interest rate $= \frac{4\%}{4} = 1\% = 0.01$.
  3. $FV = 5000(1 + 0.01)^{12} = 5000(1.01)^{12} \approx 5000(1.126825) \approx \$5634.13$.

Answer: The value is \$5,634.

Worked example 3

Binomial expansion term

Question. Find the term independent of $x$ in the expansion of $\left(x + \frac{2}{x}\right)^4$.

  1. General term: $T_{r+1} = \binom{4}{r} x^{4-r} \left(\frac{2}{x}\right)^r = \binom{4}{r} 2^r x^{4 - 2r}$.
  2. For term independent of $x$, set exponent $= 0$: $4 - 2r = 0 \implies r = 2$.
  3. $T_3 = \binom{4}{2} 2^2 = 6 \times 4 = 24$.

Answer: 24.

Before you move on

Topic 1 Revision checklist

  • Can I perform operations in standard form and round correctly to 3 significant figures?
  • Can I distinguish between arithmetic ($u_n = u_1 + (n-1)d$) and geometric ($u_n = u_1 r^{n-1}$) sequences and use their sum formulas?
  • Can I apply compound interest and depreciation formulas with various compounding periods?
  • Can I simplify expressions and solve equations using exponent and logarithm laws, while verifying domain restrictions?
  • Can I state the condition for convergence of an infinite geometric series ($|r| < 1$) and calculate $S_\infty$?
  • Can I write a direct deductive algebraic proof using representations for even, odd, and consecutive integers?
  • Can I apply the binomial general term formula $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ to find specific coefficients and terms independent of $x$?

About this guide

Independent SJMaths resource; not an official IB publication. Follows the AA syllabus first assessed in 2021 (final assessment: November 2028). Use the separate 2029 guide for the new course.

Check the IB Mathematics programme page and the official AA subject guide for the authoritative syllabus and examination-session requirements.