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IB Maths AA: Number and algebra — Practice Questions & Answers
Topic 1 builds the algebraic toolkit for the whole IB Mathematics AA course: arithmetic and geometric sequences and series, sigma notation, the sum to infinity of a convergent geometric series, the laws of exponents and logarithms, the binomial theorem, surds and indices, solving systems of linear equations, methods of proof, and (HL only) complex numbers in Cartesian, polar and exponential form. Mastering these manipulations underpins calculus, functions and statistics later in the course.
50 practice questions available for this topic — here are 10 with full answers and explanations.
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Practice questions with answers
1
Multiple choice · Easy
An arithmetic sequence has first term $u_1 = 5$ and common difference $d = 3$. What is the value of $u_{10}$?
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Why Using $u_n = u_1 + (n-1)d$, we get $u_{10} = 5 + 9 \times 3 = 32$.
2
Multiple choice · Easy
A geometric sequence has first term $u_1 = 3$ and common ratio $r = 2$. What is the value of $u_5$?
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Why Using $u_n = u_1 r^{n-1}$, we get $u_5 = 3 \times 2^{4} = 48$.
3
Multiple choice · Easy
Evaluate $\sum_{k=1}^{4} (2k+1)$.
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Why The terms are $3 + 5 + 7 + 9 = 24$.
4
Fill in the blank · Easy
An arithmetic series has first term $u_1 = 2$ and common difference $d = 2$. The sum of the first 20 terms is $S_{20} = $
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Answer:
420
Why Using $S_n = \frac{n}{2}(2u_1 + (n-1)d)$, $S_{20} = 10(4 + 38) = 420$.
5
Multiple choice · Easy
Write $\sqrt{50}$ in its simplest surd form.
$5\sqrt{2}$
$25\sqrt{2}$
$2\sqrt{5}$
$10\sqrt{5}$
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Why $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$.
6
Multiple choice · Easy
Simplify $\frac{x^{5}}{x^{2}}$.
$x^{3}$
$x^{7}$
$x^{2.5}$
$x^{10}$
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Why When dividing powers with the same base, subtract exponents: $x^{5-2} = x^{3}$.
7
Multiple choice · Easy
Evaluate $\log_{2} 32$.
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Why Since $2^{5} = 32$, $\log_{2} 32 = 5$.
8
Fill in the blank · Easy
A geometric sequence begins $4, 12, 36, \dots$ The common ratio is
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Answer:
3
Why Divide a term by the previous term: $\frac{12}{4} = 3$.
9
Multiple choice · Easy
Find the sum to infinity of a geometric series with first term $a = 8$ and common ratio $r = \frac{1}{2}$.
$16$
$\frac{16}{3}$
$4$
$8$
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Why $S_{\infty} = \frac{a}{1-r} = \frac{8}{1 - \frac{1}{2}} = 16$.
10
Multiple choice · Medium
Find the sum of the arithmetic series $7 + 11 + 15 + \dots + 47$.
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Why Here $d = 4$ and $n = \frac{47 - 7}{4} + 1 = 11$, so $S = \frac{11}{2}(7 + 47) = 297$.
Key terms in Number and algebra
Natural numbers: The counting numbers $0, 1, 2, 3, \dots$ denoted by the set $\mathbb{N}$.
Integers: The set $\mathbb{Z}$ of whole numbers including negatives, zero and positives.
Rational numbers: Numbers that can be written as a fraction $\frac{p}{q}$ with integers $p$ and $q$ and $q \neq 0$, forming the set $\mathbb{Q}$.
Irrational numbers: Real numbers that cannot be written as a ratio of two integers, such as $\sqrt{2}$ and $\pi$.
Real numbers: The set $\mathbb{R}$ containing all rational and irrational numbers.
Number set: A collection of numbers sharing a common property, such as $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ or $\mathbb{R}$.
Scientific notation: Writing a number in the form $a \times 10^{k}$ where $1 \le a \lt 10$ and $k$ is an integer.
Standard form: Another name for scientific notation, expressing very large or very small numbers compactly.
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