Number Sequence

​Number Sequences

By Victor Itumah

​0UTLINE

• What is a Sequence? Rules & Notation

• Arithmetic & Geometric Sequences

• Finding the nth Term Formula

• Special Sequences — Fibonacci & Square Numbers

• Exam-Style Questions from 11+ Papers

• Exit Ticket & Recap

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What is a Sequence?

• a sequence is "a list of things (usually numbers) that are in order."

• Each item in the sequence is called a term.

• Sequences can be infinite (go on forever) or finite

Examples of Finite and Infinite Sequences

{1, 3, 5, 7} — first 4 odd numbers

{4, 3, 2, 1} — countdown from 4

{Monday, Tuesday, ...Sunday}

These sequences have a definite end.

​Finite Sequence

{1, 2, 3, 4, ...} — counting numbers

{2, 4, 6, 8, ...} — even numbers

{1, 4, 9, 16, ...} — square numbers

The "..." means the pattern continues

forever..

​Infinite Sequence

​Key Vocabulary

​Vocabulary	​Meaning
​Term:	​each number in the sequence (e.g. the 3rd term of {3,5,7,9} is 7)
​Common Difference (d):	​the amount added or subtracted each time (arithmetic sequences)
​Common Ratio	​(r): the number multiplied each time (geometric sequences)
​nth term:	​a formula using n (position number) to find any term
​Rule:	​the instruction that connects one term to the next

ARITHMETIC & GEOMETRIC SEQUENCES

Arithmetic Sequences

— Add or subtract a fixed amount each time.

The difference between consecutive terms is always the same.
This difference is called the common difference

Worked Example:

Sequence: 3, 7, 11, 15, 19, ... Find the rule.

Differences: 7−3=4, 11−7=4, 15−11=4 → common difference = 4

It goes UP by 4 each time.

Rule (as a formula): nth term = 4n − 1

Check: n=1 → 4(1)−1=3 ■ n=2 → 4(2)−1=7 ■ n=5 → 4(5)−1=19 ■

Answer: nth term = 4n − 1

Geometric Sequences

Multiply or divide by a fixed amount each time.

Worked Example:

Sequence: 2, 6, 18, 54, ___ Find the missing term.

​Ratios: 6÷2=3, 18÷6=3, 54÷18=3 → common ratio = 3
Multiply each term by 3.
Missing term: 54 × 3 = 162
Also works backwards: 2÷3? No — always multiply FORWARDS.
Answer: 162

​Practice: Arithmetic or Geometric?

For each sequence, state
(a) the type
(b) the rule
(c) the missing term.

FINDING THE nth TERM

The nth term formula lets us find ANY term in a sequence without writing them all out.

If the sequence goes up by 2 each time, start with "2n" then adjust.

Formula for Arithmetic nth Term

Step 1: Find the common difference (d) — this becomes the coefficient of n.

Step 2: Work out what d × 1 gives, then compare to the 1st term.

Step 3: Add or subtract the difference to complete the formula.

General formula: nth term = dn + (a − d)
where a = first term, d = common difference

Worked Example: Find the nth term of: 5, 8, 11, 14, 17, ...

Step 1: Common difference = 8−5 = 3 → start with 3n

Step 2: 3×1 = 3, but first term is 5 → we need +2

Formula: nth term = 3n + 2

Verify: n=1 → 3+2=5 ■ n=4 → 12+2=14 ■ n=10 → 30+2=32

Answer: nth term = 3n + 2

Worked Example: What is the 10th term of the sequence with nth term = 5n − 3?

Substitute n = 10 into the formula:

5(10) − 3 = 50 − 3 = 47

Answer:47

SPECIAL SEQUENCES

​Some sequences have special names. Recognising them is a superpower in 11+ exams!

​Sequence Name	​Pattern	​First Terms	​nth Term
​Square Numbers	​n × n	​1, 4, 9, 16, 25, 36 ...	​n
​Cube Numbers	​n × n × n	​1, 8, 27, 64, 125 ...	​n
​Triangular Numbers	​1+2+3+...+n	​1, 3, 6, 10, 15 ...	​n(n+1)/2
​Fibonacci Numbers	​Add previous 2 terms	​1, 1, 2, 3, 5, 8, 13 ...	​No simple formula
​Powers of 2	​Double each time	​1, 2, 4, 8, 16, 32 ...	2

Worked Example:

​Graham creates a sequence with nth term 3n² + 1. What are the

first two terms? (11+)

n=1: 3(1)² + 1 = 3×1 + 1 = 4

n=2: 3(2)² + 1 = 3×4 + 1 = 13

Answer: 4, 13

■ Fibonacci Challenge!

The Fibonacci sequence adds the two previous terms:

1, 1, 2, 3, 5, 8, ?, 21

→ Missing term: 5 + 8 = 13 ( 11+)

■

Shape Patterns

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​https://www.topmarks.co.uk/ordering-and-sequencing/shape-patterns

EXAM-STYLE QUESTIONS

​Now try these questions — just like in real 11+ papers. Identify the pattern, write the rule, then find the answer.

​#	​Question	​Answer	​Explanation
​1.	​What is the rule for: 4, 11, 18, 25, 32, 39,	​Add 7	​Difference = 11−4 = 7 each time
​2.	​Missing: 5, 10, 20, ___, 80, 160	​40	​×2 each time; 20×2=40
​3.	​Next two terms: 1, 4, 9, 16, 25, ___, __	​36, 49	​Square numbers: 6
​4.	​nth term of: 5, 8, 11, 14, 17, ...	​3n+2	​d=3; 3(1)+2=5
​5.	​8th term in: 6, 10, 14, 18, 22, ...	​34	​nth term=4n+2; 4(8)+2=34
​6.	​Missing: 393, 384, 375, 366, 357, ___	​348	​−9 each time; 357−9=348
​7.	​Missing: 0.5, 1.0, 2.0, 4.0, ___, 16.0	​8.0	​×2 each time; 4.0×2=8.0

■ Quick-Fire Practice

EXIT TICKET & RECAP

Open Ended

1. What is the difference between an arithmetic and a geometric sequence?

Open Ended

2. Write the nth term for: 7, 10, 13, 16, 19,...

Open Ended

3. The nth term of a sequence is 4n − 1. What is the 7th term?

​Key Takeaways

• Arithmetic sequence: add or subtract a fixed number each time (common difference).

• Geometric sequence: multiply or divide by a fixed number each time (common ratio).

• nth term formula: dn + (a−d) for arithmetic sequences, where d = common difference, a = first term.

• Special sequences: square, cube, triangular, Fibonacci — know them by heart!