E2.7 Sequences

Learning objectives

E2.7.1Continue sequences• Continue a given number sequence or pattern; subscript notation may be used.E2.7.2Recognise sequence patterns• Recognise term-to-term rules and relationships in linear, quadratic, cubic and exponential sequences and simple combinations of them.E2.7.3Find and use the nth term• Find and use the nth term of sequences.

To continue a sequence, identify a rule that works between every displayed pair of consecutive terms. The position of a term may be written with subscript notation such as $u_1,u_2,u_3$ .

Pattern check	What to calculate	Typical continuation
additive	first differences	add or subtract the same amount
multiplicative	ratios of consecutive non-zero terms	multiply or divide by the same factor
alternating or cyclic	separate odd/even positions or repeating operations	repeat the full cycle
changing differences	differences, then second or third differences	extend the difference pattern first

For $6,13,32,69,130,\ldots$ , the terms match $n^3+5$ : $1^3+5=6$ , $2^3+5=13$ , and so on. The next term is $6^3+5=221$ .

For $100,50,25,12.5,6.25,\ldots$ , each term is half the previous one, so the next term is $3.125$ .

A rule must fit all shown transitions. Do not assume a constant difference after checking only the first pair, and do not confuse the term value $u_n$ with its position $n$ .

The way differences or ratios behave reveals a sequence family. Recognising the family narrows the possible term-to-term and position-to-term rules.

Sequence family	Diagnostic pattern	Common nth-term shape
linear	constant first difference	$an+b$
quadratic	constant second difference	$an^2+bn+c$
cubic	constant third difference	$an^3+bn^2+cn+d$
exponential	constant non-zero ratio	$ar^{n-1}$

$4,9,14,19,\ldots$ is linear because first differences are $5$ . $3,10,29,66,\ldots$ is cubic because it matches $n^3+2$ . $1,4,16,64,\ldots$ is exponential with ratio $4$ .

A simple combination such as $n^3+2^n$ may not show an immediately constant difference or ratio. Compare the terms with familiar powers, subtract the identifiable component, and test the remaining pattern.

A curved growth pattern is not automatically exponential. Use constant differences or ratios as evidence, and verify the proposed relationship against every given term.

An nth-term rule gives the value at position $n$ directly. Its algebraic form should match the recognised sequence family and reproduce every supplied term.

Family	Starting move
linear	constant difference $a$ gives $an+b$ ; use one term to find $b$
quadratic	constant second difference is $2a$ ; subtract $an^2$ and find the remaining linear rule
cubic	constant third difference is $6a$ ; subtract $an^3$ and analyse the remainder
exponential	ratio $r$ gives $ar^{n-1}$ , where $a$ is the first term

$4,9,14,19,\ldots:\quad u_n=5n-1$

Check positions $n=1,2,3$ before accepting a rule. For $u_n=5n-1$ , these give $4,9,14$ , matching the sequence.

To find a term, substitute its positive integer position. To decide whether $331$ belongs to $u_n=5n-1$ , solve $5n-1=331$ : $n=66.4$ , not a positive integer, so $331$ is not a term.

For $24,12,6,3,\ldots$ , $u_n=24(1/2)^{n-1}$ . The exponent is $n-1$ so that $u_1=24$ .

Matching only the next term does not prove an nth-term rule. Verify all displayed terms, and when testing membership require $n$ to be a permitted positive integer.

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—E2.8 ProportionRead

E2.7 Sequences

Learning objectives

Continue a sequence by identifying its rule

Recognise linear, quadratic, cubic and exponential patterns

Find and use an nth-term rule

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