Sequences
Section: Algebra and graphs 1 | Syllabus: Cambridge IGCSE Mathematics (0580)
Describing and Continuing Sequences
A sequence is a list of numbers or terms arranged in a particular order, following a rule. The term-to-term rule describes how to get from one term to the next.
- A sequence can go up (increasing) or down (decreasing), and can change by addition, subtraction, multiplication or division
- The term-to-term rule is the operation applied to one term to generate the next
- To continue a sequence, first find the term-to-term rule, then apply it again
Worked Example: Finding a Term-to-Term Rule
- Question: Write down the term-to-term rule for this sequence.
2, 10, 50, 250- Step 1: Divide consecutive terms to test for a common ratio: 102=5, 5010=5, 25050=5
- Step 2: Each term is 5 times the one before it
- Answer: Multiply by 5
Worked Example: Continuing a Linear Sequence
- Question: These are the first four terms of a different sequence.
41, 35, 29, 23
Find the next two terms in the sequence.- Step 1: Find the difference between consecutive terms: 35-41=-6, 29-35=-6, 23-29=-6
- Step 2: The term-to-term rule is subtract 6
- Answer: 17, 11
Common Mistakes
MistakeAssuming every sequence increases, and writing a negative common difference as positive
Fixalways subtract a term from the one directly after it (later − earlier) to get the correctly signed difference
MistakeTesting only the first pair of terms for the rule and assuming it holds throughout
Fixcheck at least two or three consecutive pairs before deciding on the term-to-term rule
Finding the nth Term of a Linear Sequence
The nth term is a formula, in terms of n, that generates any term of the sequence directly - without listing every term before it.
- For a linear sequence, the nth term has the form dn + c, where d is the common difference
- Find c by comparing the sequence with the multiples of d - it is what must be added to reach the first term
- The nth term can be used to find any term without listing the sequence, or to test whether a given number belongs to it
Worked Example: Deriving the nth Term
- Question: These are the first four terms of a sequence.
9, 13, 17, 21
Find an expression for the nth term.- Step 1: Common difference: 13-9=4
- Step 2: Compare with the 4 times table (4, 8, 12, 16) - each term is 5 more than this
- Answer: 4n + 5
Worked Example: Testing Whether a Number Is a Term
- Question: The nth term of a sequence is 6n + 1. Is 100 a term of this sequence?
- Step 1: Set the nth term equal to 100: 6n+1=100
- Step 2: Solve for n: n=16.5
- Answer: No - n is not a whole number, so 100 is not a term
Common Mistakes
MistakeUsing the common difference alone as the constant, e.g. writing 4n + 4 instead of 4n + 5 for 9, 13, 17, 21
Fixsubstitute n = 1 into the finished formula and check it gives the first term of the sequence
MistakeConcluding a number is in the sequence just because the equation dn + c = value can be solved
Fixalways check that the solution for n is a positive whole number before accepting it
Special Sequences
Certain sequences appear so often in mathematics that they have their own names and are worth recognising on sight.
- Square numbers: 1, 4, 9, 16, 25, ... formed by n^2
- Cube numbers: 1, 8, 27, 64, 125, ... formed by n^3
- Triangular numbers: 1, 3, 6, 10, 15, ... each one adds one more than the last; nth term (n(n+1)/2)
- A Fibonacci-type sequence is generated by adding the two previous terms to find the next one
Worked Example: Continuing a Fibonacci-Type Sequence
- Question: A sequence starts 2, 5, 7, 12, 19, ... . Each term after the second is found by adding the two terms before it. Find the next term.
- Step 1: Check the rule: 2+5=7, 5+7=12, 7+12=19
- Step 2: Add the last two terms: 12+19=31
- Answer: 31
Common Mistakes
MistakeThinking the nth term of the triangular numbers is n(n + 1), without dividing by 2
Fixremember the division by 2 - the triangular-number formula is n(n + 1) ÷ 2
MistakeConfusing square numbers with doubling, e.g. thinking the 4th square number is 4 × 2 = 8
Fixa square number is n multiplied by itself, not by 2 - the 4th square number is 4 × 4 = 16
Sequences of Patterns and Diagrams
Sequences are often given as a series of diagrams built from shapes or tiles rather than as a plain list of numbers. Counting each diagram and organising the results in a table reveals the pattern.
- Count the number of each type of tile or shape in the first few diagrams and record the results in a table
- Look separately at how each part of the pattern (e.g. grey tiles, white tiles) grows from diagram to diagram
- A quantity that grows by a constant amount each time gives a linear expression in n; a quantity where the differences themselves grow gives a more complex expression
Worked Example: Finding Formulas from a Tile Pattern
- Question: A sequence of patterns is made from grey and white tiles.
Diagram 1 has 3 grey tiles and 0 white tiles.
Diagram 2 has 6 grey tiles and 3 white tiles.
Diagram 3 has 9 grey tiles and 8 white tiles.
Find expressions, in terms of n, for the number of grey tiles and the number of white tiles in diagram n.- Step 1: Grey tiles: 3, 6, 9, ... increase by 3 each time, so the formula is 3n
- Step 2: White tiles: 0, 3, 8, ... the differences are 3 and 5 - not constant, so test n^2-1: n=10,\ n=23,\ n=38 - all match
- Answer: grey = 3n, white = n² − 1
Common Mistakes
MistakeAssuming every diagram sequence must be linear, and forcing a constant-difference rule onto a quadratic pattern
Fixcheck whether the differences between terms are themselves constant - if not, try n² or a similar expression as a starting point
MistakeMiscounting tiles in the diagrams, especially later diagrams that are drawn smaller or more crowded
Fixcount systematically, row by row or shape by shape, rather than all at once
Quadratic Sequences
Extended Only
In a quadratic sequence, the differences between consecutive terms are not constant, but the second differences (the differences between the differences) are.
- A quadratic sequence has an nth term of the form an^2+bn+c
- The second difference is always equal to 2a, so a is found by halving it
- Once a is known, subtract an^2 from the original sequence to leave a linear sequence for the remaining part
Worked Example: Finding the nth Term of a Quadratic Sequence
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