🔢 Sequences

Grade 8 · Cambridge Lower Secondary Stage 8

What you'll learn

Find the nth term of arithmetic (linear) sequences
Find the nth term of geometric sequences
Use the nth term formula to find specific terms
Determine if a value is a term in a sequence
Recognise and work with special sequences

📖 Learn

1. Arithmetic Sequences

Each term increases/decreases by a constant common difference d.

nth term = a + (n−1)d where a = first term, d = common difference

3, 7, 11, 15, ... → d = 4, a = 3 → nth term = 3 + (n−1)×4 = 4n − 1

20, 17, 14, 11, ... → d = −3, a = 20 → nth term = 20 + (n−1)(−3) = 23 − 3n

💡 Shortcut: the nth term = d×n + c where c = a − d

2. Finding the nth Term Formula

Step 1: Find d (common difference between consecutive terms).
Step 2: Write dn. Step 3: Adjust the constant by comparing to the sequence.

Sequence: 5, 8, 11, 14 → d = 3 → 3n → check: 3(1)=3 but T₁=5, so add 2 → 3n + 2

Sequence: 1, 3, 5, 7 → d = 2 → 2n → check: 2(1)=2 but T₁=1, so subtract 1 → 2n − 1

3. Geometric Sequences

Each term is multiplied by a constant common ratio r.

nth term = a × r^(n−1)

2, 6, 18, 54 → r = 3 → nth term = 2 × 3^(n−1)

80, 40, 20, 10 → r = ½ → nth term = 80 × (½)^(n−1)

4. Is n a Term in the Sequence?

Set the nth term formula equal to the value and solve for n. If n is a positive integer, it's a term.

nth term = 4n − 1. Is 59 a term? → 4n − 1 = 59 → n = 15 ✓ (yes, it's the 15th term)

Is 62 a term? → 4n − 1 = 62 → n = 15.75 ✗ (not an integer, so no)

5. Special Sequences

Memorise these:

Square numbers: 1, 4, 9, 16, 25 ... → n²

Cube numbers: 1, 8, 27, 64 ... → n³

Triangular numbers: 1, 3, 6, 10, 15 ... → n(n+1)/2

Fibonacci: 1, 1, 2, 3, 5, 8, 13 ... each term = sum of previous two

✏️ Worked Examples

Example 1 — Find the nth term

Find the nth term of 7, 11, 15, 19, ...

Step 1: d = 11 − 7 = 4

Step 2: nth term = 4n + c. When n=1: 4(1)+c = 7 → c = 3

Result: 4n + 3

Check: n=4 → 4(4)+3 = 19 ✅

Example 2 — Find a Specific Term

The nth term of a sequence is 5n − 2. Find the 20th term.

T₂₀ = 5(20) − 2 = 100 − 2 = 98

Example 3 — Geometric Sequence

Find the 6th term of 3, 6, 12, 24, ...

Step 1: r = 6/3 = 2

Step 2: T₆ = 3 × 2^(6−1) = 3 × 32 = 96

Example 4 — Is it in the Sequence?

nth term = 3n + 1. Is 100 a term?

3n + 1 = 100 → 3n = 99 → n = 33 ✓ integer → Yes, 100 is the 33rd term.

🎨 Visualizer

🔢 Sequence Generator

Enter a first term and common difference to generate a sequence.

First term (a): Difference (d): Terms:

📊 Sequence Visualiser

Using values from generator above.

🔍 nth Term Finder

Enter 4 consecutive terms and find the nth term formula.

🎯 Sequence Spotter

Exercise 1 — Next Terms

Find the next term in each sequence.

Exercise 2 — Finding the nth Term Formula

Enter the coefficient of n (i.e. d) in the nth term formula.

Exercise 3 — Using the nth Term

Use the given formula to find the specified term.

Exercise 4 — Geometric Sequences

Find the next term or specified term.

Exercise 5 — Is it a Term? & Special Sequences