Number and Algebra — IB Math AA SL AA SL Study Guide

For: IB Math AA SL candidates sitting IB Math: Analysis & Approaches SL.

Covers: Arithmetic and geometric sequences, logarithm laws and equations, binomial theorem for positive integer exponents, exponential equations, and sigma notation as outlined in the IB Math AA SL Number and Algebra syllabus.

You should already know: IGCSE / pre-DP math.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the IB Math AA SL style for educational use. They are not reproductions of past IBO papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official IBO mark schemes for grading conventions.

1. What Is Number and Algebra?

Number and Algebra is the foundational branch of IB Math AA SL focused on quantifiable patterns, variable relationships, and systematic algebraic manipulation. It accounts for 20-25% of your final AA SL exam mark, making it one of the highest-yield topics in the syllabus. All subsequent topics (including calculus, statistics, and functions) rely on the core rules and formulas covered in this guide, so mastery here directly boosts your performance across the entire course.

2. Sequences — arithmetic and geometric

A sequence is an ordered list of terms that follow a consistent pattern. The two most commonly tested sequence types in IB Math AA SL are arithmetic and geometric sequences.

Arithmetic sequences

Arithmetic sequences have a constant common difference $d$ between consecutive terms. For an arithmetic sequence with first term $u_1$, the $n$-th term is calculated as: $$u_n=u_1+(n-1)d$$ The sum of the first $n$ terms of an arithmetic sequence can be written in two equivalent forms, depending on the values you are given: $$S_n=\frac{n}{2}(u_1+u_n)=\frac{n}{2}\left(2u_1+(n-1)d\right)$$ Worked example: An arithmetic sequence has first term $u_1=5$ and common difference $d=3$. Find the 12th term and the sum of the first 12 terms.

1. $u_{12}=5+(12-1)(3)=5+33=38$
2. $S_{12}=\frac{12}{2}(5+38)=6\ast 43=258$

Geometric sequences

Geometric sequences have a constant common ratio $r$ between consecutive terms. For a geometric sequence with first term $u_1$, the $n$-th term is: $$u_n=u_1{r}^{n-1}$$ The sum of the first $n$ terms of a geometric sequence is: $$S_n=\frac{u_1(1-r^n)}{1-r}=\frac{u_1(r^n-1)}{r-1}\text{for }r\ne 1$$ If $|r|<1$, the sequence converges, and you can calculate the sum of an infinite number of terms: $$S_{\infty }=\frac{u_1}{1-r}$$ Exam tip: Examiners frequently include trick questions where $|r|\ge 1$; never use the infinite sum formula without first confirming the ratio is between -1 and 1.

3. Logarithm laws and equations

Logarithms are the inverse operation of exponents, defined by the relationship: $$a^x=b⟺{\log }_{a}b=x$$ where $a>0,a\ne 1$, and $b>0$ (the argument of a logarithm must always be positive, a rule that trips up many students on exams).

The three core logarithm laws you will use to simplify and solve equations are:

1. Product rule: ${\log }_a(xy)={\log }_{a}x+{\log }_{a}y$
2. Quotient rule: ${\log }_a\left(\frac{x}{y}\right)={\log }_{a}x-{\log }_{a}y$
3. Power rule: ${\log }_a{x}^k=k{\log }_{a}x$

You will also use the change of base formula to calculate logarithms with non-standard bases using your calculator: $${\log }_{a}b=\frac{\ln b}{\ln a}=\frac{{\log }_{10}b}{{\log }_{10}a}$$

Worked example: Solve ${\log }_3(x+4)+{\log }_3(x-2)=3$

1. Combine the logs using the product rule: ${\log }_3\left[(x+4)(x-2)\right]=3$
2. Convert to exponential form: $(x+4)(x-2)=3^3=27$
3. Expand and rearrange: $x^2+2x-8=27⟹x^2+2x-35=0$
4. Solve the quadratic: $x=5$ or $x=-7$
5. Check validity: $x=-7$ makes the argument of the second log negative, so discard it. Final solution: $x=5$

4. Binomial theorem (positive integer exponent)

The binomial theorem lets you expand expressions of the form $(a+b{)}^n$ where $n$ is a positive integer, without multiplying out each term manually. The coefficient of each term is given by the binomial coefficient: $$\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}$$ where $n!$ (n factorial) is the product of all positive integers up to $n$.

The full binomial expansion is written as: $$(a+b{)}^n=\sum _{r=0}^n(\genfrac{}{}{0ex}{}{n}{r})a^{n-r}{b}^r$$

Worked example: Find the coefficient of $x^2$ in the expansion of $(3x+4{)}^5$

1. The general term in the expansion is $\left(\genfrac{}{}{0ex}{}{5}{r}\right)(3x{)}^{5-r}(4{)}^r$
2. We need the power of $x$ to be 2, so $5-r=2⟹r=3$
3. Calculate the coefficient: $\left(\genfrac{}{}{0ex}{}{5}{3}\right)(3{)}^2(4{)}^3=10\ast 9\ast 64=5760$ Exam tip: You do not need to expand the entire expression when asked for a single specific term; only calculate the term corresponding to the required value of $r$ to save time.

5. Exponential equations

Exponential equations are equations where the unknown variable appears in the exponent. There are two common cases you will encounter on exams:

1. Same base on both sides: If $a^x=a^y$ and $a\ne 1$, then $x=y$. Simply equate the exponents and solve for the unknown. Worked example: Solve $2^{3x-2}=16$. Rewrite 16 as $2^4$, so $3x-2=4⟹3x=6⟹x=2$.
2. Different bases on both sides: Take the natural logarithm (or base 10 logarithm) of both sides, then use the logarithm power rule to bring the exponent down and isolate the unknown. Worked example: Solve $4^{x+1}=6^x$, giving your answer to 3 significant figures.
3. Take natural log of both sides: $\ln (4^{x+1})=\ln (6^x)$
4. Apply power rule: $(x+1)\ln 4=x\ln 6$
5. Rearrange to isolate $x$: $x\ln 4+\ln 4=x\ln 6⟹\ln 4=x(\ln 6-\ln 4)$
6. Calculate: $x=\frac{\ln 4}{\ln 6-\ln 4}\approx \frac{1.3863}{1.7918-1.3863}\approx 3.42$ Exam tip: IB requires all non-exact final answers to be given to 3 significant figures. Do not round intermediate steps, as this will introduce rounding error; keep values stored in your calculator until the final step.

6. Sigma notation

Sigma notation ($\sum$) is a shorthand way to write the sum of a sequence of terms. The standard form is: $$\sum _{r=a}^{b}f(r)=f(a)+f(a+1)+f(a+2)+...+f(b)$$ where $r$ is the index of summation, $a$ is the lower limit of the sum, and $b$ is the upper limit of the sum. The index is a "dummy variable", so ${\sum }_{r=1}^{5}r={\sum }_{k=1}^{5}k=1+2+3+4+5=15$.

Worked example: Calculate ${\sum }_{r=2}^5(r^2-1)$

1. Substitute each value of $r$ from 2 to 5 into the expression:

• $r=2$: $4-1=3$
• $r=3$: $9-1=8$
• $r=4$: $16-1=15$
• $r=5$: $25-1=24$

2. Add the terms: $3+8+15+24=50$ Sigma notation is frequently used to write the sum of arithmetic and geometric sequences, as you saw earlier in the binomial theorem formula.

7. Common Pitfalls (and how to avoid them)

• Pitfall 1: Keeping invalid logarithm solutions with negative arguments. Why it happens: You focus on solving the algebraic equation and forget the domain constraint for logarithms. Correct move: Always plug every solution back into the original logarithm expressions to confirm all arguments are positive; discard any invalid solutions.
• Pitfall 2: Using the infinite geometric sum formula when $|r|\ge 1$. Why it happens: You mix up the conditions for finite and infinite geometric sums. Correct move: Before calculating $S_{\infty }$, first confirm the ratio $r$ is strictly between -1 and 1. If $|r|\ge 1$, the infinite sum does not exist.
• Pitfall 3: Off-by-one errors in sequence term calculations. Why it happens: You use $n$ instead of $n-1$ in the $n$-th term formula for arithmetic or geometric sequences. Correct move: Test your formula with $n=1$ to confirm it gives the correct first term before solving for larger values of $n$.
• Pitfall 4: Miscalculating binomial coefficients by mixing up $n$ and $r$. Why it happens: You forget that the exponent of $b$ in the binomial term matches the lower value in the binomial coefficient $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$. Correct move: When asked for the term with $x^k$, set $r=k$ and calculate $\left(\genfrac{}{}{0ex}{}{n}{k}\right)a^{n-k}{b}^k$.
• Pitfall 5: Rounding intermediate steps when solving exponential equations. Why it happens: You write down rounded values instead of storing them in your calculator. Correct move: Keep all intermediate values in your calculator memory, only round the final answer to 3 significant figures.

8. Practice Questions (IB Math AA SL Style)

Question 1

A geometric sequence has first term $u_1=16$ and third term $u_3=4$. (a) Find the two possible values of the common ratio $r$. (b) For the value of $r$ that gives a converging sequence, find the infinite sum $S_{\infty }$.

Solution

(a) Using the geometric nth term formula: $u_3=u_1{r}^2$ $$4=16r^2⟹r^2=\frac{1}{4}⟹r=\frac{1}{2}\text{ or }r=-\frac{1}{2}$$ (b) Both values of $r$ have $|r|<1$, so both give converging sequences. We will use the positive ratio for this example: $$S_{\infty }=\frac{16}{1-\frac{1}{2}}=32$$ For $r=-\frac{1}{2}$, $S_{\infty }=\frac{16}{1-(-\frac{1}{2})}=\frac{32}{3}\approx 10.7$

Question 2

Expand $(2x-1{)}^4$ fully using the binomial theorem.

Solution

Apply the binomial expansion formula for $n=4$: $$\begin{align*} (2x - 1)^4 &= \binom{4}{0}(2x)^4(-1)^0 + \binom{4}{1}(2x)^3(-1)^1 + \binom{4}{2}(2x)^2(-1)^2 + \binom{4}{3}(2x)^1(-1)^3 + \binom{4}{4}(2x)^0(-1)^4 \ &= 116x^41 + 48x^3(-1) + 64x^21 + 42x(-1) + 111 \ &= 16x^4 - 32x^3 + 24x^2 - 8x + 1 \end{align*}$$

Question 3

Solve ${\log }_2(x)+{\log }_2(x+2)=3$, justifying why you discard any invalid solutions.

Solution

1. Combine logs using the product rule: ${\log }_2\left[x(x+2)\right]=3$
2. Convert to exponential form: $x(x+2)=8$
3. Rearrange to quadratic: $x^2+2x-8=0$
4. Solve: $x=2$ or $x=-4$
5. Justification: $x=-4$ is invalid because the argument of ${\log }_2(x)$ would be negative, which is undefined. Final solution: $x=2$

9. Quick Reference Cheatsheet

10. What's Next

Mastery of Number and Algebra is non-negotiable for success in the rest of the IB Math AA SL syllabus, as these concepts underpin every subsequent topic. For example, geometric sequences are used to model continuous growth and decay in calculus, the binomial theorem is used to derive binomial probability distributions in statistics, and logarithm and exponential rules are core to solving differentiation and integration problems for non-polynomial functions. You will also encounter sigma notation extensively when learning about Riemann sums for integration later in the course.

To reinforce your understanding of these topics, work through official past paper questions aligned with the Number and Algebra syllabus, and cross-check your answers against official IBO mark schemes to get familiar with grading expectations. If you get stuck on any problem, or need further clarification on any rule or formula, you can ask Ollie for step-by-step help anytime on the homepage.