Number and algebra practice questions (IB Maths AA)

Sample questions

All 2 free IB Maths AA questions on this topic, from the free IB Maths AA SL Paper 1 Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 16 marks · Arithmetic sequences & series
   An arithmetic sequence has third term $u_3 = 11$ and seventh term $u_7 = 23$.
   (a)

   Find the first term $u_1$ and the common difference $d$.

   [4]
   (b)

   Find $S_{20}$, the sum of the first 20 terms.

   [2]
   Show answerFrom the free IB Maths AA SL Paper 1 Practice Exam →
2. Question 26 marks · Exponents & logarithms
   This question is about exponents and logarithms.
   (a)

   Solve $3^{2x - 1} = 27^{x - 2}$.

   [3]
   (b)

   Show that $2\log_3 6 - \log_3 4 = 2$.

   [3]
   Show answerFrom the free IB Maths AA SL Paper 1 Practice Exam →

Key terms

The definitions and formulas IB Maths AA expects for number and algebra, typeset as they appear on the paper.

Arithmetic sequence and series

A sequence with a common difference $d$. The $n$th term and the sum of the first $n$ terms are in the formula booklet; the second sum form is useful when the last term is known.

$$u_n = u_1 + (n-1)d, \qquad S_n = \frac{n}{2}\big(2u_1 + (n-1)d\big) = \frac{n}{2}(u_1 + u_n)$$

Geometric sequence and series

A sequence with a common ratio $r$. The sum to infinity exists only when $|r| < 1$, a condition examiners expect you to state.

$$u_n = u_1 r^{n-1}, \qquad S_n = \frac{u_1(r^n - 1)}{r - 1}, \qquad S_\infty = \frac{u_1}{1 - r}$$

Laws of logarithms

The rules for combining and splitting logs of the same base, plus the change-of-base formula. Solving $a^x = b$ gives $x = \log_a b$.

$$\log_a xy = \log_a x + \log_a y, \quad \log_a \frac{x}{y} = \log_a x - \log_a y, \quad \log_a x^n = n\log_a x, \quad \log_b a = \frac{\log_c a}{\log_c b}$$

Binomial theorem

The expansion of $(a + b)^n$ for a positive integer $n$ using binomial coefficients, which can be read from Pascal's triangle or computed from factorials.

$$(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r, \qquad \binom{n}{r} = \frac{n!}{r!\,(n-r)!}$$

Compound interest

The future value of an amount $PV$ invested at $r$ percent per year, compounded $k$ times a year for $n$ years. A Paper 2 question usually expects the GDC finance solver.

$$FV = PV\left(1 + \frac{r}{100k}\right)^{kn}$$

Complex numbers (HL)

Numbers of the form $z = a + bi$ with modulus $|z| = \sqrt{a^2 + b^2}$ and argument $\theta$. In polar or Euler form powers and roots follow de Moivre's theorem.

$$z = r e^{i\theta} = r(\cos\theta + i\sin\theta), \qquad z^n = r^n(\cos n\theta + i\sin n\theta)$$

Proof by mathematical induction (HL)

Show the statement holds for the base case, assume it holds for $n = k$, show it then holds for $n = k + 1$, and conclude. The written conclusion is itself worth a mark.

Frequently asked questions

Other IB Maths AA topics

• Functions
• Geometry and trigonometry
• Statistics and probability
• Calculus

All IB Maths AA topics · IB Maths AA exam guide · All free practice exams