Sequences: including other sequences including where r is a surd (Higher tier only)

Overview

A sequence is an ordered list of terms. In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio, r. At Higher tier, that multiplier can be a surd, such as \sqrt{2} or \sqrt{3}. The sequence 1,\sqrt{2},2,2\sqrt{2},4 therefore follows a simple rule despite its unfamiliar appearance.

This guide targets AQA GCSE Mathematics 8300, section 3.2.4, A24: the Higher-only extension to other sequences, including geometric sequences where r is a surd. Generating terms links to A23. General sequence recognition is not all Higher-only, but this surd-ratio extension is. 1

These questions bring together exact arithmetic, simplifying surds, index laws and algebraic reasoning. Candidates may be asked to continue a sequence, identify its multiplier, work backwards, calculate a stated term or explain whether a claim is correct. AQA can assess this content on any of its three papers. 2

Your aim is to recognise the structure and show enough working for a marker to follow it. The questions and allocations in this guide are original AQA-style teaching examples, not official past-paper questions or guaranteed future mark schemes. Practise exact answers first, then tackle missing terms and unfamiliar rules.

Key Concepts

1. Recognise a geometric sequence by its multiplier

The defining feature is a constant ratio, not a constant difference. For 2,2\sqrt{3},6, calculate

\frac{2\sqrt{3}}{2}=\sqrt{3},\qquad \frac{6}{2\sqrt{3}}=\sqrt{3}.

For the second calculation, check that 2\sqrt{3}\times\sqrt{3}=6. Both adjacent ratios agree. The common ratio is \sqrt{3}, so the next term is 6\sqrt{3}.

Compare this with an arithmetic sequence, where the same number is added each time. A geometric sequence multiplies by the same number. Writing “add \sqrt{3}” describes the wrong operation, even if your first calculation happens to look plausible.

Check more than one adjacent pair when recognising a pattern. However, finitely many displayed terms do not determine a unique infinite sequence: use the sequence type or rule stated in the question. Credit given for an explanation should rest on calculations, not “it looks geometric”.

Cover and recall: What operation finds r, and which way round does the division go? Answer: later term divided by the immediately preceding term.

2. Understand why surds must stay exact

A surd is an irrational root written in exact form. For example, \sqrt{2} cannot be written as a fraction of two integers. By contrast, \sqrt{4}=2, so it is not a surd. Square roots of positive non-square integers, such as \sqrt{5}, are the main examples here.

Replacing \sqrt{2} by 1.4 changes the number. In particular, 1.4^2=1.96, whereas (\sqrt{2})^2=2 exactly. Repeated decimal rounding compounds the error. Leave roots in the calculation until a decimal approximation is explicitly required.

An exact answer need not contain a root sign. In 1,\sqrt{2},2,2\sqrt{2},4, the third and fifth terms simplify to integers. The irrational multiplier does not make every term irrational.

Why does this work? A square root is defined so that squaring it returns its original non-negative radicand. That is why two matching square roots simplify so cleanly.

3. Simplify products before moving on

For non-negative u and v,

\sqrt{u}\sqrt{v}=\sqrt{uv}.

In particular, \sqrt{k}\sqrt{k}=k. Multiply any outside coefficients separately:

2\sqrt{3}\times3\sqrt{3}=2\times3\times3=18.

Also remove square factors: \sqrt{12}=\sqrt{4\times3}=2\sqrt{3}. If the multiplier is presented as \sqrt{8}, simplify it to 2\sqrt{2} before using it. Then its square is 8, not 2.

Do not apply multiplication rules to addition. For example, \sqrt{2}+\sqrt{2}=2\sqrt{2}, not 2. AQA expects simplified answers where appropriate; an instruction to simplify is not always needed. 1

Examiner checkpoint: \sqrt{4} is exact but not fully simplified. Write 2 on the answer line. In a question requesting simplest form, finish this final step.

4. Continue the sequence one multiplication at a time

For 1,\sqrt{2},2,2\sqrt{2},4, each arrow means multiply by \sqrt{2}. Continue:

4\times\sqrt{2}=4\sqrt{2},\qquad 4\sqrt{2}\times\sqrt{2}=8.

The diagram also shows a useful shortcut. Two successive multiplications by \sqrt{2} equal one multiplication by 2. Therefore the odd-position terms are 1,2,4,8,\ldots, and the even-position terms are \sqrt{2},2\sqrt{2},4\sqrt{2},\ldots.

This alternation is specific to the starting value and multiplier. If the first term is \sqrt{3} and the ratio is \sqrt{2}, the first terms are \sqrt{3},\sqrt{6},2\sqrt{3},2\sqrt{6}. They are all irrational. Avoid claiming that every surd-ratio sequence alternates between whole numbers and surds.

Memory hook: “Pair the roots; release the whole.” Two factors of \sqrt{2} become 2; two factors of \sqrt{3} become 3.

5. Find a named term without listing everything

A useful position-to-term formula is

t_n=ar^{n-1}.

Here t_n means the term at position n, a is the first term and r is the common ratio. The position n is a positive integer. This formula is a derived tool for geometric sequences, not the separate A25 requirement about linear and quadratic nth terms.

Why n-1 rather than n? The first term is already present: t_1=a. Term 2 needs one multiplication, term 3 needs two, and term 6 needs five. Checking n=1 gives ar^0=a for a non-zero ratio.

If a=3 and r=\sqrt{2}, then

t_6=3(\sqrt{2})^5=3(2\times2\times\sqrt{2})=12\sqrt{2}.

Keep the multiplier in brackets when raising it to a power. If r=2\sqrt{2}, then r^3=(2\sqrt{2})^3=16\sqrt{2}. Writing 2(\sqrt{2})^3 would miss two of the coefficient factors.

Cover and recall: Explain the exponent without quoting the formula. Answer: count the multiplications from the first position to the required position.

6. Work backwards and handle separated terms

Going forwards multiplies by r; going backwards divides by r. If t_3=18 and r=\sqrt{3},

t_2=\frac{18}{\sqrt{3}}=6\sqrt{3},\qquad t_1=\frac{6\sqrt{3}}{\sqrt{3}}=6.

You can also recognise that two forward steps multiply by (\sqrt{3})^2=3, giving t_1=18/3=6 directly.

When given t_3=18 and t_5=54, the quotient 54/18=3 equals r^2, because the terms are two positions apart. It does not equal r. If the ratio is stated to be positive, r=\sqrt{3}. Without that condition, both signs satisfy r^2=3, so use the rest of the information to decide.

In general, for non-zero terms and positive integer gap k,

\frac{t_{n+k}}{t_n}=r^k.

Candidates gain useful method evidence by writing the term gap before choosing the power.

7. Recognise decreasing and sign-changing examples

A surd ratio can be smaller than one. With a=8 and r=1/\sqrt{2}=\sqrt{2}/2, the terms are

8,\ 4\sqrt{2},\ 4,\ 2\sqrt{2},\ 2.

Positive starting terms decrease when 0<r<1. Rationalising a denominator gives an equivalent exact form: 1/\sqrt{2}=\sqrt{2}/2. Do not turn the multiplier into \sqrt{2}, which would make the sequence grow instead.

If a question supplies r=-\sqrt{2}, the signs alternate: 1,-\sqrt{2},2,-2\sqrt{2},4. Treat this as an extension when the rule is supplied. The root simplification remains the same, but signs must be tracked as well.

8. Distinguish geometric, quadratic and other recursive rules

Arithmetic sequences have constant first differences. Genuine quadratic sequences have constant non-zero second differences. Geometric sequences have constant adjacent ratios. A complicated-looking root does not determine the sequence type.

For example, 1,3,6,10,15 has first differences 2,3,4,5 and second differences 1,1,1. Its adjacent ratios are not constant. By contrast, 1,\sqrt{2},2,2\sqrt{2},4 has a constant multiplier.

Other recursive sequences are defined in the question under AQA's A24 notes. 1 If u_1=1 and u_{n+1}=\sqrt{2}u_n+1, then u_2=\sqrt{2}+1 and u_3=3+\sqrt{2}. The extra addition means it is not a geometric sequence, so do not use ar^{n-1}.

What if? Remove the +1. The resulting rule u_{n+1}=\sqrt{2}u_n is geometric. The operation, not the appearance of the terms, decides the classification.

Mathematical Relationships

Formulas to learn or derive

The 2027 AQA Higher formula sheet does not provide the following sequence, surd or index rules. Each is labelled Must memorise, meaning learn it or be able to derive it reliably. 3

Must memorise: t_{n+1}=rt_n. Use this term-to-term rule to generate the next term.

Must memorise: r=t_{n+1}/t_n, provided $t_n
e0$. Divide adjacent terms to find or check the multiplier.

Must memorise: t_n=ar^{n-1}. Use this derived formula to calculate a stated position.

Must memorise: t_{n-1}=t_n/r, for $r
e0, andt_{n+k}=t_nr^k$. These are the same forward/backward rules applied over one or several steps.

Must memorise: \sqrt{u}\sqrt{v}=\sqrt{uv} for u,v\ge0, and \sqrt{u}/\sqrt{v}=\sqrt{u/v} for u\ge0,v>0. A product with coefficients satisfies (b\sqrt{u})(c\sqrt{v})=bc\sqrt{uv}.

Must memorise: $\sqrt{q^2d}=|q|\sqrt{d}$ for $d\ge0$. With positive integer $q$, this becomes $q\sqrt{d}$. Use it to remove square factors.

Must memorise: (\sqrt{d})^{2m}=d^m and (\sqrt{d})^{2m+1}=d^m\sqrt{d} for positive d and non-negative integer m. These shortcuts pair the roots.

Must memorise: x^px^q=x^{p+q}, (x^p)^q=x^{pq}, (xy)^p=x^py^p and x^0=1, using integer exponents and non-zero bases where needed. They explain the power manipulation above.

Must memorise: 1/\sqrt{d}=\sqrt{d}/d for d>0. Multiply numerator and denominator by \sqrt{d} to rationalise.

Given on formula sheet, synoptic connection only: compound interest uses P(1+i/100)^N, where P is the initial amount, i the percentage rate per period and N the number of periods. 3 It illustrates repeated multiplication. Unlike the sequence formula, period counting begins from an initial amount at time zero. Do not confuse N periods with the term position n.

Geometric-series sums and logarithms are not needed for this guide's A24 skill.

Practical Applications and Graph Skills

A square with side 1 cm has diagonal \sqrt{2} cm by Pythagoras. Make that diagonal the side of a new square, and repeat. The side lengths form 1,\sqrt{2},2,2\sqrt{2},4,\ldots cm. Every two steps double the side length; every single step doubles the area. This explains why a surd multiplier can have a simple geometric meaning.

Use the same unit throughout. A ratio of two lengths is dimensionless. If one length is in metres and another in centimetres, convert before dividing: 1 m equals 100 cm. Pure number-sequence answers have no units. There is no required practical for this mathematics topic.

If asked to plot a sequence, place the position n on the horizontal axis and term value t_n on the vertical axis. For a=1,r=\sqrt{2}, the first points are (1,1),(2,\sqrt{2}),(3,2),(4,2\sqrt{2}),(5,4). Use suitable decimal approximations to locate irrational heights, but keep exact values in algebraic answers. Sequence positions are discrete, so do not automatically join the points. A sketched exponential function is a related but different object.

Exam Technique and Active Revision

Each paper allows 90 minutes for 80 marks, averaging 67.5 seconds per mark. Aim for about one minute per mark while leaving checking time; this is a revision strategy, not an AQA per-question limit. Paper 1 is non-calculator; Papers 2 and 3 allow calculators. 2

Identify the rule, count any position gap, write the calculation, then simplify. Show that requires the steps that reach the printed result. Explain needs a mathematical reason. When evaluating a method or claim, identify the error or justify the claim with evidence, not a generic discussion of advantages and disadvantages. 4

AQA's published marking glossary distinguishes method marks, accuracy marks and independent marks. Valid alternative methods can receive credit, but the detailed allocation depends on the individual question. A bare correct answer is not sufficient when working is explicitly required. 4 5

Cover each worked solution and attempt it from scratch. Then check the multiplier, exponent and exact answer form. Revisit a fresh example tomorrow and again a few days later. If you made an error, name it precisely: adding instead of multiplying, counting n jumps, or rounding too soon. Explaining that error is more useful than copying the solution again.

Listen and Recall

The ten-minute female-tutor episode follows five sections: introduction, core concepts, exam advice, a quick-fire quiz and a closing summary. Answer aloud before each quiz solution. Pause the player if you need more thinking time.

References