Cambridge Lower Secondary · Mathematics 0862 · Stage 8 · Algebra

Sequences and the nth Term

Use term-to-term rules with whole numbers, negatives and fractions, build sequences from shape patterns, and find and use the nth term an + b of a linear sequence.

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In the curriculum

Understand term-to-term rules (add, subtract, multiply or divide by a number — including fractions and negative numbers), use them to continue sequences, and make sequences from number patterns and from shape (spatial) patterns. [8As.01]
Understand and describe the nth term (position-to-term) rule of a sequence using algebra, in the forms n ± a, a × n or an ± b, where a and b are positive or negative whole numbers or fractions. Use it to find any term, find the rule for a linear sequence and decide whether a number is in a sequence. [8As.02]

A sequence is a list of numbers (or shapes) in order that follow a rule. In Stage 8 you work with two kinds of rule. A term-to-term rule tells you how to get from one term to the next. A position-to-term rule — the nth term — tells you how to work out any term straight from its position, so you can find the 100th term without writing out the first 99. This page shows you both, with fractions and negative numbers, and links them to shape patterns.

1. Key words

Word / phrase	Meaning / use
sequence	متتابعة — numbers or shapes in order, following a rule
term	حد — each number in a sequence
position, n	الترتيب — the place of a term: 1st, 2nd, 3rd … (n = 1, 2, 3 …)
term-to-term rule	قاعدة من حد للي بعده — how to get the next term, e.g. "add 4"
nth term / position-to-term rule	الحد النوني — a formula that gives the term from its position n, e.g. 4n + 1
linear sequence	متتابعة خطية — the terms go up or down by the same amount each time
common difference	الفرق الثابت — the amount added each time (can be negative or a fraction)
ascending / descending	تصاعدية / تنازلية — going up / going down
spatial pattern	نمط من الأشكال — a sequence of drawings, e.g. squares made from sticks
infinite / finite	غير منتهية / منتهية — goes on for ever / stops

2. Term-to-term rules

To describe a sequence with a term-to-term rule, give the first term and the rule. The rule can add, subtract, multiply or divide, and the numbers can be negative or fractions.

Sequence	First term	Term-to-term rule	Next two terms
5, 9, 13, 17, …	5	add 4	21, 25
40, 33, 26, 19, …	40	subtract 7	12, 5
−11, −8, −5, −2, …	−11	add 3	1, 4
3, 6, 12, 24, …	3	multiply by 2	48, 96
800, 400, 200, 100, …	800	divide by 2	50, 25
1/4, 3/4, 1 1/4, 1 3/4, …	1/4	add 1/2	2 1/4, 2 3/4
2 1/3, 2, 1 2/3, 1 1/3, …	2 1/3	subtract 1/3	1, 2/3
1, 3, 7, 15, …	1	multiply by 2, then add 1	31, 63

Finding missing terms. In a linear sequence the steps are equal. For 7, __, __, 19 there are 3 equal steps from 7 to 19, so each step is (19 − 7) ÷ 3 = 4. The missing terms are 11 and 15.

Fractions with different denominators. Write the fractions with a common denominator first. For 1/2, 5/6, 7/6, 3/2: in sixths the terms are 3/6, 5/6, 7/6, 9/6, so the rule is "add 2/6", which is "add 1/3".

3. Sequences from shape patterns

Many sequences come from drawings. Count the parts in each pattern, write the numbers as a sequence, then look at how the shape grows. The way the shape grows explains the rule.

Squares made from sticks. Each new square needs 3 more sticks.

The numbers of sticks are 4, 7, 10, … The term-to-term rule is "add 3", because each new square joins on to the one before and only needs 3 new sticks (a top, a bottom and a right side). Pattern 1 needs one extra stick for the left side. So pattern n needs 3n + 1 sticks: 3 for each of the n squares, plus 1 for the very first side.

4. The nth term (position-to-term rule)

The nth term is a formula in n. Put n = 1 to get the 1st term, n = 2 to get the 2nd term, and so on. You can jump straight to any term.

nth term	n = 1	n = 2	n = 3	n = 4	100th term
n + 6	7	8	9	10	106
n − 4	−3	−2	−1	0	96
5n	5	10	15	20	500
4n − 1	3	7	11	15	399
−2n + 10	8	6	4	2	−190
(1/2)n + 3	3 1/2	4	4 1/2	5	53

Look at the pattern: in every linear sequence the number multiplying n is the common difference. 4n − 1 goes up in 4s; −2n + 10 goes down in 2s; (1/2)n + 3 goes up in halves.

Term-to-term or position-to-term?

For 3, 7, 11, 15, … the term-to-term rule is "add 4" (it links a term to the next one). The position-to-term rule is 4n − 1 (it links a term to its position n). Read the question carefully to see which one it asks for.

5. Finding the nth term of a linear sequence

Find the common difference d (the amount added each time). It may be negative or a fraction.
Write dn — the sequence of multiples of d.
Find the zero term: go one step backwards from the 1st term (zero term = 1st term − d).
nth term = dn + zero term.
Check with the 2nd or 3rd term.

Going one step back from the first term gives the zero term.

Sequence	Difference d	Zero term	nth term	Check (n = 3)
7, 12, 17, 22, …	5	7 − 5 = 2	5n + 2	15 + 2 = 17 ✓
20, 17, 14, 11, …	−3	20 − (−3) = 23	−3n + 23 (or 23 − 3n)	−9 + 23 = 14 ✓
−6, −2, 2, 6, …	4	−6 − 4 = −10	4n − 10	12 − 10 = 2 ✓
2 1/2, 3, 3 1/2, 4, …	1/2	2 1/2 − 1/2 = 2	(1/2)n + 2	1 1/2 + 2 = 3 1/2 ✓
3/4, 1 1/4, 1 3/4, 2 1/4, …	1/2	3/4 − 1/2 = 1/4	(1/2)n + 1/4	1 1/2 + 1/4 = 1 3/4 ✓
8, 9, 10, 11, …	1	8 − 1 = 7	n + 7	3 + 7 = 10 ✓

Only linear sequences have an nth term of the form an + b. For 3, 6, 12, 24, … the differences are 3, 6, 12 — not equal — so it is not linear, and an nth term such as 3n does not work. Describe it with its term-to-term rule ("multiply by 2") instead.

6. Is a number in the sequence?

Set the nth term equal to the number and solve. If n is a positive whole number, the number is in the sequence (and n tells you its position). If n is not a whole number, it is not in the sequence.

Is 100 a term of 3n + 1? 3n + 1 = 100 → 3n = 99 → n = 33. Yes — it is the 33rd term.
Is 150 a term of 3n + 1? 3n + 1 = 150 → 3n = 149 → n = 49.66… Not a whole number, so no. (Also: the terms of 3n + 1 are always 1 more than a multiple of 3, and 150 is a multiple of 3.)

الفرق بين القاعدتين

قاعدة حد لحد: بتقولك إزاي تروح من حد للي بعده، زي "زوّد 5".
الحد النوني: بيديك أي حد على طول من ترتيبه n. الفرق الثابت هو اللي بيتضرب في n، وبعدين ارجع خطوة لورا من الحد الأول عشان تعرف تزود أو تطرح كام (الحد الصفري).
مثال: 7، 12، 17 → الفرق 5 والحد الصفري 2 → الحد النوني 5n + 2.

7. Worked examples

Worked example 1 — The first term of a sequence is 2 3/4. The term-to-term rule is "subtract 1/2". (a) Write the next four terms. [2] (b) Write down the first negative term. [1]

Part	Working and answer
(a)	2 3/4 − 1/2 = 2 1/4; then 1 3/4; then 1 1/4; then 3/4. Answer: 2 1/4, 1 3/4, 1 1/4, 3/4
(b)	Keep going: 3/4 → 1/4 → 1/4 − 1/2 = −1/4. Answer: −1/4

Common mistake: 1/4 − 1/2 is not 0 and not 1/4. Change to quarters: 1/4 − 2/4 = −1/4.

Worked example 2 — These are the first five terms of a linear sequence: 4, __, __, 25, __. Find the missing terms. [2]

From 4 (1st term) to 25 (4th term) there are 3 equal steps. Each step = (25 − 4) ÷ 3 = 21 ÷ 3 = 7. Answer: 4, 11, 18, 25, 32. Common mistake: dividing by 4 (the number of terms) instead of 3 (the number of steps).

Worked example 3 — The nth term of a sequence is 6n − 4. (a) Write down the first three terms. [1] (b) Work out the 50th term. [1] (c) Show that 100 is not a term of this sequence. [2]

Part	Working and answer
(a)	n = 1: 6 − 4 = 2; n = 2: 12 − 4 = 8; n = 3: 18 − 4 = 14. 2, 8, 14
(b)	6 × 50 − 4 = 300 − 4 = 296
(c)	6n − 4 = 100 → 6n = 104 → n = 17.33… This is not a whole number, so 100 is not a term. (Check: the 17th term is 98 and the 18th term is 104.)

Worked example 4 — Find an expression for the nth term of the sequence 9, 5, 1, −3, … [2]

The difference is 5 − 9 = −4, so start with −4n.
Zero term = 9 − (−4) = 13.
nth term = −4n + 13 (or 13 − 4n).
Check n = 4: −16 + 13 = −3 ✓

Common mistake: writing 4n + 13 (the sequence goes DOWN, so the coefficient of n is negative) or −4n + 9 (using the first term instead of the zero term).

Worked example 5 — Look at the stick patterns of squares in section 3. (a) How many sticks are in Pattern 10? [1] (b) Write an expression for the number of sticks in Pattern n. [1] (c) Explain, using the diagram, why your expression works. [1] (d) Can a pattern in this sequence use exactly 85 sticks? Show how you decide. [1]

Part	Working and answer
(a)	Sticks: 4, 7, 10, … add 3 each time. 3 × 10 + 1 = 31 sticks
(b)	Difference 3, zero term 4 − 3 = 1 → 3n + 1
(c)	Each square adds 3 new sticks (top, bottom, right side), and there is 1 extra stick on the far left: 3 × n + 1.
(d)	3n + 1 = 85 → 3n = 84 → n = 28, a whole number. Yes — Pattern 28.

Worked example 6 — A sequence starts 1 1/4, 1 1/2, 1 3/4, 2, … (a) Find the nth term. [2] (b) Which term is equal to 10? [1]

Part	Working and answer
(a)	Difference = 1/4. Zero term = 1 1/4 − 1/4 = 1. nth term = (1/4)n + 1 (or n/4 + 1). Check n = 4: 1 + 1 = 2 ✓
(b)	(1/4)n + 1 = 10 → (1/4)n = 9 → n = 36. The 36th term.

Common mistakes

Giving the term-to-term rule when the nth term is asked for: for 7, 12, 17, … the answer "n + 5" is wrong; the nth term is 5n + 2.
Using the first term instead of the zero term: 7, 12, 17 is not 5n + 7. Check your rule with n = 1.
Losing the negative sign: 20, 17, 14, … has nth term −3n + 23, not 3n + 23.
Slips with fractions and mixed numbers — change to a common denominator before adding or subtracting.
Saying a number IS in the sequence because n = 49.7 "rounds to 50". n must be a whole number.
Using an nth term like an + b for a sequence that is not linear (e.g. 3, 6, 12, 24).
Dividing by the number of terms instead of the number of steps when filling gaps.

How Checkpoint asks this

"Write down the next term" and "Write down the term-to-term rule" — usually 1 mark each. Say the operation AND the number ("subtract 3/4", not just "3/4").
"Complete the table" of pattern number and number of sticks/tiles — then "Find an expression for the number of … in pattern n".
"Find the nth term" — always check your answer by substituting n = 1 and n = 2.
"Show that 150 is not in the sequence" — solve the equation and say clearly that n is not a whole number.
"Draw a ring around the nth term of this sequence" or "Ali says … Is he correct? Explain." — test each option with n = 1, 2, 3.
Diagram questions: "Explain how you know" — link the numbers to the picture (what each new pattern adds).

Practice

Remember

Term-to-term rule = first term + what you do each time (add, subtract, multiply, divide — fractions and negatives allowed).
nth term of a linear sequence = (difference) × n + (zero term). Zero term = 1st term − difference.
A falling sequence has a negative coefficient of n, e.g. 23 − 3n.
To test whether a number is a term, solve nth term = number; it is a term only if n is a positive whole number.
In shape patterns, explain the rule from the picture: what each new pattern adds, plus what the first one needs extra.

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