C2.7 Sequences

Learning objectives

C2.7.1Continue a given number sequence or pattern.• Continue a given number sequence or pattern.C2.7.2Recognise patterns in sequences, including the term-to-term rule, and• Recognise patterns in sequences, including the term-to-term rule, and relationships between different sequences.C2.7.3Find and use the nth term of the following sequences: (a) linear (b)• Find and use the nth term of the following sequences: (a) linear (b) simple quadratic (c) simple cubic. e.g. write the next two terms in this sequence: 1, 3, 6, 10, 15, …, … e.g. find the nth term of 2, 5, 10, 17

To continue a sequence, identify the operation that connects consecutive terms and check that it works across every given step before applying it again.

Given terms	Changes	Continuation
$7,13,19,25$	add 6 each time	$31$
$3,9,27,81$	multiply by 3 each time	$243$
$27,26,23,18,11$	subtract $1,3,5,7$	subtract 9, then 11: $2,-9$
$18,21,26,33,42$	add $3,5,7,9$	add 11: $53$

Write the first differences beneath the terms. If they are constant, repeat that difference. If they form a simple sequence themselves, continue the difference pattern first and then use the new difference on the last term.

Differences are not always the controlling pattern. For $18,20,24,32,48$ , the added amounts double: $+2,+4,+8,+16$ , so add 32 next to obtain 80.

Use at least three transitions to test a proposed rule. A rule that fits only the final pair may not explain the whole sequence; after predicting a term, verify that its change continues the same pattern.

Continue only the stated or strongly supported pattern. A short finite list can fit many invented rules, so prefer the simplest consistent rule signalled by all given terms; finding a general nth term is handled in the later card.

A term-to-term rule tells how to obtain the next term from the current one. A pattern table may contain several linked sequences, so compare both changes within each row and relationships between rows.

Evidence	Likely rule to test
constant first differences	add or subtract a fixed amount
constant ratio	multiply or divide by a fixed amount
differences follow a pattern	continue the difference sequence
two rows change together	test a sum, difference, multiple or square relationship

For $30,26,22,18$ , each term is 4 less than the previous one, so the term-to-term rule is 'subtract 4'. State the operation and amount, not merely the next term.

A rule can be reversed. If 'multiply by 3, then subtract 1' produces the fourth term 68, undo subtraction first and multiplication second: $(68+1)/3=23$ , so the third term is 23.

In a pattern table, track each row separately before comparing rows. If small-square counts are $2,4,6,8$ while dot counts are $6,9,12,15$ , their term-to-term rules are add 2 and add 3; the rows share the same diagram number but not the same change.

A term-to-term rule uses the preceding term; an nth-term rule uses the position number directly. Do not call 'add 6' the nth term, and do not assume two linked rows must have identical differences.

An nth-term formula gives a term directly from its position $n$ . Constant first, second or third differences reveal whether a sequence is linear, quadratic or cubic.

Sequence type	Constant layer	Leading coefficient	Starting model
linear	first difference $d$	$d$	$dn+b$
quadratic	second difference $D_2$	$a=D_2/2$	$an^2+bn+c$
cubic	third difference $D_3$	$a=D_3/6$	$an^3+bn^2+cn+d$

For $12,19,26,33,40$ , the first difference is 7. Start with $7n$ ; its first term is 7, which is 5 too small, so the nth term is $7n+5$ .

For $2,5,10,17$ , first differences are $3,5,7$ and the second difference is 2, so the leading term is $n^2$ . Since $n^2$ gives $1,4,9,16$ , add 1 to every term: the nth term is $n^2+1$ .

For $3,17,55,129,251$ , the constant third difference is 12, so the leading term is $(12/6)n^3=2n^3$ . Comparing with $2,16,54,128,250$ shows a constant remainder 1, giving $2n^3+1$ .

To generate terms, substitute $n=1,2,3,\ldots$ . To find a position, set the formula equal to the target and solve for a positive integer $n$ . For $7n+5=40$ , $n=5$ , so 40 is the fifth term.

Differences diagnose the degree; do not use the linear shortcut when first differences change. Sequence positions begin at $n=1$ unless stated otherwise, and a non-integer or non-positive solution for $n$ is not a valid term position.

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C2.7 Sequences

Learning objectives

Continue sequences from their changing pattern

Recognise term-to-term rules and linked sequence patterns

Find and use linear, quadratic and cubic nth terms

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