Module 3 of 3Lesson 1 of 11 · Unit 23 · Sequences and the nth term
Mathematics Form 2 · Term 3 · Unit 23

🔤 Sequences and the nth term

AlgebraIGCSE path ✓JC ✓7As.017As.028As.018As.029As.019As.02
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Why this matters

Patterns are everywhere: seats in a stadium, tiles on a floor, the price of a taxi ride as the kilometres tick up. If you can find the rule behind a pattern, you can jump straight to the 100th step without drawing the 99 before it. That rule is called the nth term, and it is one of the most powerful ideas in algebra.

🎯 By the end of this unit

  • I can continue a sequence and describe its term-to-term rule.
  • I can find and use the nth term of a linear sequence in the form an + b.
  • I can decide whether a number is in a sequence.
  • I can find sequences from patterns and real-life situations.
  • I can recognise square numbers and simple sequences based on n².
📖 Key words

sequence
a list of numbers in order that follows a rule
term
each number in a sequence
position (n)
where a term is in the sequence: 1st, 2nd, 3rd, …
term-to-term rule
how to get from one term to the next, such as “add 3”
nth term
a rule that gives any term from its position n, such as 3n + 2
linear sequence
a sequence that goes up or down by the same amount each time
difference
the amount added (or subtracted) each time
square numbers
1, 4, 9, 16, …: the numbers n²

1 Learn it

Patterns make sequences

Look at these matchstick squares. Each new square needs 3 more matches, because it shares one side with the square before.

So the numbers of matches make the sequence 4, 7, 10, 13, … Each number is a term. The term-to-term rule is “add 3”.

Pattern 14 matchesPattern 27 matchesPattern 310 matches…
4, 7, 10, … matches: each pattern adds 3.
⭐ Key ideaA term-to-term rule tells you the NEXT term. It is slow if you want the 100th term, because you need all the terms before it.

Position-to-term: the nth term

Put the terms in a table with their position n.

The sequence 5, 8, 11, 14 goes up in 3s, just like the 3 times table (3, 6, 9, 12). Each term is 2 more than the 3 times table.

So each term is 3 × n + 2. We write the nth term as 3n + 2.

Position n1234
3n36912
Term581114
n = 15n = 28n = 311n = 4143 × n (the 3 times table)+ 2 extra every time
The blue blocks are the 3 times table; the 2 orange blocks are the same in every term.
⭐ Key ideaDifference 3 → the rule starts with 3n. Then adjust: what do you add to 3n to get the term?

Finding the nth term quickly

Step 1: find the difference, a. The nth term starts with an.

Step 2: step back one place from the first term to find the zero term, b. The nth term is an + b.

For 7, 11, 15, 19: the difference is 4, and stepping back from 7 gives 7 − 4 = 3.

  1. Difference: 4, so start with 4n
  2. Zero term: \(7 – 4 = 3\)
  3. nth term \(= 4n + 3\)
  4. Check n = 2: 4 × 2 + 3 = 11 ✓
3n = 0+47n = 1+411n = 2+415n = 3+419n = 4zero termdifference 4 → 4n; zero term 3 → 4n + 3
Stepping back from the 1st term gives the zero term, the number you add.
⭐ Key ideaGoing DOWN? Then the difference is negative: 20, 17, 14, … has difference −3 and zero term 23, so the nth term is 23 − 3n.

Using the nth term

To find any term, put its position in place of n.

To test whether a number is in the sequence, set the nth term equal to it and solve. If n comes out as a whole number, the number is in the sequence. If not, it is not.

  1. 50th term of \(4n + 3\): 4 × 50 + 3 = 203
  2. Is 95 in \(4n + 3\)? Solve 4n + 3 = 95
  3. \(4n = 92\), so \(n = 23\): yes, the 23rd term
  4. Is 100 in \(4n + 3\)? \(4n = 97\), so n = 24.25: no
⭐ Key ideaThere is no 24.25th term. Positions are always whole numbers 1, 2, 3, …

Sequences in real life

Tables in a restaurant are pushed together in a row. Each extra table adds 2 more chairs, one on each long side.

Real-life rules are often linear: a fixed start-up amount plus the same amount for each step. A taxi that charges P15 to start and P6 per kilometre costs 6n + 15 pula for n km.

1 table4 chairs2 tables6 chairs3 tables8 chairs
4, 6, 8, … chairs: the nth term is 2n + 2.
⭐ Key ideaMatch the parts to the story: the number in front of n is “how much each step adds”; the number on its own is the fixed part.

Square numbers: a sequence that is not linear

Square numbers come from square patterns of dots: 1, 4, 9, 16, 25, … The nth term is \(n^2\).

The differences are 3, 5, 7, 9, …, so they are not the same each time. This is not a linear sequence.

Many sequences are built from n²: for example \(n^2 + 1\) gives 2, 5, 10, 17, … and \(2n^2\) gives 2, 8, 18, 32, …

1² = 12² = 43² = 94² = 16each new square adds an L-shape of 3, 5, 7, … dots
Each square number is the previous one plus an L-shape of an odd number of dots.
⭐ Key ideaIf the differences are not the same, the sequence is not linear. Compare it with the square numbers 1, 4, 9, 16, 25.

2 Watch it work

EXAMPLE 1

Find and use the nth term

For the sequence 7, 11, 15, 19, … find (a) the nth term (b) the 20th term.
  1. (a) Difference: \(11 – 7 = 4\), so start with 4n
  2. Zero term: \(7 – 4 = 3\)
  3. nth term \(= 4n + 3\)
  4. (b) \(4 \times 20 + 3\)
  5. \(= 83\)
Answer (a) 4n + 3 (b) 83
EXAMPLE 2

A decreasing sequence

For 20, 17, 14, 11, … (a) find the nth term (b) is −10 in the sequence?
  1. (a) Difference: 17 − 20 = −3, so start with −3n
  2. Zero term: \(20 + 3 = 23\)
  3. nth term \(= 23 – 3n\)
  4. (b) Solve 23 − 3n = −10
  5. \(33 = 3n\)
  6. \(n = 11\), a whole number
Answer (a) 23 − 3n (b) Yes, −10 is the 11th term
EXAMPLE 3

Is the number in the sequence?

The nth term of a sequence is 3n + 2. Is 100 in the sequence?
  1. Solve 3n + 2 = 100
  2. \(3n = 98\)
  3. n = 32.66…
  4. n is not a whole number
  5. Check: the 32nd term is 98 and the 33rd term is 101
Answer No, 100 is not in the sequence
EXAMPLE 4

A matchstick pattern

Look at the matchstick squares in Learn it (4, 7, 10, … matches). (a) Find a formula for the number of matches in Pattern n. (b) How many matches are needed for 30 squares? (c) What is the longest row of squares you can make with 200 matches?
  1. (a) Difference 3, zero term 4 − 3 = 1, so 3n + 1
  2. (b) 3 × 30 + 1 = 91
  3. (c) Solve 3n + 1 = 200
  4. \(3n = 199\), so n = 66.33…
  5. 66 squares use 3 × 66 + 1 = 199 matches; 67 squares would need 202
Answer (a) 3n + 1 (b) 91 matches (c) 66 squares
EXAMPLE 5

A sequence built on square numbers

Find the nth term of 2, 5, 10, 17, 26, … and the 10th term.
  1. Differences: 3, 5, 7, 9, which are not equal, so it is not linear
  2. Compare with the square numbers: 1, 4, 9, 16, 25
  3. Each term is 1 more than a square number
  4. nth term \(= n^2 + 1\)
  5. 10th term: \(10^2 + 1 = 101\)
Answer \(n^2 + 1\); the 10th term is 101

3 Your turn

Try each question in your book first. Then tap the button to check every step.

1🟢 Warm-up
Write down the next two terms and the term-to-term rule: 6, 11, 16, 21, …
👀 Show the answer
Step by step
  1. The terms go up by 5 each time
  2. \(21 + 5 = 26\)
  3. \(26 + 5 = 31\)
Answer 26, 31; add 5
2🟢 Warm-up
Write down the first four terms of the sequence with nth term 2n + 5.
👀 Show the answer
Step by step
  1. n = 1: \(2 + 5 = 7\)
  2. n = 2: \(4 + 5 = 9\)
  3. n = 3: \(6 + 5 = 11\)
  4. n = 4: \(8 + 5 = 13\)
Answer 7, 9, 11, 13
3🟢 Warm-up
Find the nth term of 4, 8, 12, 16, …
👀 Show the answer
Step by step
  1. These are the multiples of 4
  2. Difference 4 and zero term 0
Answer 4n
4🟢 Warm-up
Write down the first five square numbers. What is the 12th square number?
👀 Show the answer
Step by step
  1. \(1^2, 2^2, 3^2, 4^2, 5^2\)
  2. 1, 4, 9, 16, 25
  3. \(12^2 = 144\)
Answer 1, 4, 9, 16, 25; 144
5🔵 Core
Find the nth term of 9, 15, 21, 27, …
👀 Show the answer
Step by step
  1. Difference: 6, so start with 6n
  2. Zero term: \(9 – 6 = 3\)
Answer 6n + 3
6🔵 Core
Find the nth term of 50, 46, 42, 38, …
👀 Show the answer
Step by step
  1. Difference: −4, so start with −4n
  2. Zero term: \(50 + 4 = 54\)
Answer 54 − 4n
7🔵 Core
Find the nth term of \(\dfrac{1}{2}, 1, 1\dfrac{1}{2}, 2, …\) and write down its 20th term.
👀 Show the answer
Step by step
  1. The terms go up by \(\dfrac{1}{2}\) each time, so start with \(\dfrac{1}{2}n\)
  2. Zero term: \(\dfrac{1}{2} – \dfrac{1}{2} = 0\)
  3. nth term \(= \dfrac{1}{2}n\), which is the same as \(\dfrac{n}{2}\)
  4. 20th term: \(\dfrac{20}{2} = 10\)
Answer nth term \(\dfrac{n}{2}\); 20th term 10
8🔵 Core
The nth term of a sequence is 7n − 3. Find the 100th term.
👀 Show the answer
Step by step
  1. \(7 \times 100 – 3\)
  2. \(= 697\)
Answer 697
9🔵 Core
(a) Is 75 a term of the sequence 4n + 3? (b) Is 150 a term of 6n + 5?
👀 Show the answer
Step by step
  1. (a) \(4n + 3 = 75\), so \(4n = 72\)
  2. \(n = 18\), a whole number: yes
  3. (b) 6n + 5 = 150, so \(6n = 145\)
  4. n = 24.16…, not a whole number: no
Answer (a) Yes, the 18th term (b) No
10🔵 Core
A taxi in Gaborone charges P15 to start plus P6 for every kilometre. (a) Write a formula for the cost, in pula, of a journey of n km. (b) Find the cost of a 12 km journey.
👀 Show the answer
Step by step
  1. (a) Each km adds P6, and P15 is fixed
  2. Cost \(= 6n + 15\)
  3. (b) \(6 \times 12 + 15\)
  4. \(= 87\)
Answer (a) 6n + 15 (b) P87.00
11🔵 Core
Tables are pushed together in a row, with chairs around them. (a) Find the nth term for the number of chairs. (b) How many chairs fit around 20 tables?
1 table4 chairs2 tables6 chairs3 tables8 chairs
👀 Show the answer
Step by step
  1. Chairs: 4, 6, 8, … so the difference is 2
  2. Zero term: \(4 – 2 = 2\)
  3. (a) nth term \(= 2n + 2\)
  4. (b) 2 × 20 + 2 = 42
Answer (a) 2n + 2 (b) 42 chairs
12🟠 Stretch
A linear sequence has 3rd term 17 and 7th term 37. Find its first term and its nth term.
👀 Show the answer
Step by step
  1. From the 3rd term to the 7th term is 4 steps
  2. The terms went up by 37 − 17 = 20
  3. Difference: \(20 \div 4 = 5\)
  4. 1st term: 17 − 2 × 5 = 7
  5. Zero term: \(7 – 5 = 2\)
  6. nth term \(= 5n + 2\)
Answer First term 7; nth term 5n + 2
13🟠 Stretch
Which is the first term of the sequence 3n + 4 that is bigger than 500?
👀 Show the answer
Step by step
  1. Solve 3n + 4 = 500
  2. \(3n = 496\), so n = 165.33…
  3. So the 165th term is just below 500
  4. The 166th term: 3 × 166 + 4 = 502
Answer The 166th term, which is 502
14🟠 Stretch
Find the nth term of 3, 6, 11, 18, 27, … and its 10th term.
👀 Show the answer
Step by step
  1. Differences: 3, 5, 7, 9, not equal, so not linear
  2. Square numbers: 1, 4, 9, 16, 25
  3. Each term is 2 more than a square number
  4. nth term \(= n^2 + 2\)
  5. 10th term: \(10^2 + 2 = 102\)
Answer \(n^2 + 2\); 102
15🟠 Stretch
Neo’s sequence has nth term 5n + 1. Lesego’s has nth term 3n + 11. At which position are their terms equal, and what is the term?
👀 Show the answer
Step by step
  1. Set them equal: 5n + 1 = 3n + 11
  2. \(2n = 10\)
  3. \(n = 5\)
  4. Term: 5 × 5 + 1 = 26
  5. Check: 3 × 5 + 11 = 26 ✓
Answer The 5th terms are both 26

4 Watch out!

✗ Wrong3, 7, 11, 15, … goes up in 4s, so the nth term is n + 4.
✓ RightThe nth term is \(4n – 1\).

The difference goes in FRONT of n: “up in 4s” means 4n. Then adjust: 4n gives 4, 8, 12, …, which is 1 too big, so 4n − 1.

✗ WrongThe nth term of 5, 8, 11, 14 is “+3”.
✓ RightThe term-to-term rule is +3. The nth term is 3n + 2.

“+3” only tells you the next term. The nth term must give any term from its position n.

✗ Wrong20, 17, 14, … : nth term \(3n + 23\)
✓ Rightnth term \(23 – 3n\)

The sequence goes DOWN by 3, so the difference is −3. Check n = 1: 23 − 3 = 20 ✓, but 3 + 23 = 26 ✗.

✗ Wrong3n + 2 = 100 gives \(n = 32.7\), so 100 is about the 33rd term.
✓ Rightn is not a whole number, so 100 is NOT in the sequence.

Positions are whole numbers. The 32nd term is 98 and the 33rd is 101; 100 is skipped.

5 In your exams

🇧🇼 Junior Certificate (JCE)

In JCE Paper 1, sequences usually appear as multiple choice: the next term, a missing term, or which expression is the nth term. In Paper 2 (often Section B or C), you meet a pattern of matchsticks, dots or tiles: you complete a table, find the nth term, use it for a large pattern number, and decide whether a given number of sticks is possible. Always check your nth term with the first two terms.

🎓 Later in IGCSE

IGCSE 0580 Core asks for the nth term of linear sequences and simple ones based on n², often from a diagram pattern. Extended papers go further: quadratic and cubic sequences, exponential sequences such as \(2^n\), and sequences with fractions.

🏆 Challenge
Pattern n is a square of n × n white tiles with a border of grey tiles all around it. (a) Find a formula for the number of grey tiles in Pattern n. (b) Find a formula for the total number of tiles. (c) Which pattern uses 144 tiles in total, and how many of them are grey?
Pattern 1Pattern 2Pattern 3
👀 Reveal the solution
Step by step
  1. Grey tiles: 8, 12, 16, … so difference 4 and zero term 8 − 4 = 4
  2. (a) Grey tiles \(= 4n + 4\)
  3. The whole pattern is a square with side n + 2
  4. (b) Total tiles \(= (n + 2)^2\)
  5. (c) \((n + 2)^2 = 144\), so n + 2 = 12
  6. \(n = 10\)
  7. Grey: 4 × 10 + 4 = 44
  8. Check: \(100\) white \(+ 44\) grey = 144 ✓
Answer (a) 4n + 4 (b) \((n + 2)^2\) (c) Pattern 10, with 44 grey tiles
📝 Remember

  • The term-to-term rule gives the next term; the nth term gives ANY term from its position n.
  • Linear sequence nth term = an + b: a is the difference, b is the zero term (one step before the 1st term).
  • Decreasing sequences have a negative difference, e.g. 23 − 3n.
  • A number is in the sequence only if solving an + b = number gives a whole-number n.
  • If the differences are not equal, compare with the square numbers 1, 4, 9, 16, 25 (nth term n²).

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