A sequence is a list of numbers that follow a pattern, like 5, 8, 11, 14, … Think of laying paving slabs in a row, adding the same number each time. A linear sequence goes up (or down) by the same amount at every step — that fixed step is the common difference.
Why this matters: patterns like these describe anything that changes at a steady rate — saving the same amount each week, seats added row by row in a theatre, the cost of a taxi as the miles tick up. Finding a rule (the nth term) means you can jump straight to the 100th value without writing out all 99 before it.
A linear sequence goes up by the same amount (+3) each time
The nth term — a formula for the whole sequence
Each term adds 3 amber dots — the +3 step becomes the 3n in the rule.
Rather than listing terms forever, we find a rule (the nth term) that gives any term you want — feed in the position n, out comes that term. Here n = 1 is the first term, n = 2 the second, and so on.
The number in front of n is the common difference. Then adjust by a fixed number so the first term comes out right.
5, 8, 11, 14, … • It goes up by 3 each time, so start with 3n.
• 3 × 1 = 3, but the first term is 5 — that's 2 more. So the rule is 3n + 2.
Check: n = 2 gives 3(2) + 2 = 8 ✓
Position n
1
2
3
4
3n
3
6
9
12
3n + 2 (term)
5
8
11
14
The 3n bar sets the step; the +2 block shifts it so term 1 is 5.
Sequences can also go down. For 20, 17, 14, 11, … the difference is −3, so the rule starts with −3n; adjusting gives 23 − 3n (the same as −3n + 23).
Is a number in the sequence?
Set the nth term equal to the number and solve for n. If n turns out to be a positive whole number, the number is in the sequence and n tells you its position; if not, it isn't there.
Is 45 in 4n + 1? 4n + 1 = 45 → 4n = 44 → n = 11. A whole number, so yes — it's the 11th term.
Solve for n: a whole number means it's a term; a decimal means it isn't.
Is 100 in 3n + 2? 3n + 2 = 100 → 3n = 98 → n = 32.67. Not a whole number, so no.
Common mistake: the rule is 3n + 2, not n + 3. The "+3" is the gap between terms; the number that actually multiplies n is the common difference. Always test your rule on term 1 and term 2 before trusting it.
In a nutshell: the common difference is the number in front of n; finish by adjusting so term 1 is correct. To test whether a value appears, solve for n and check it's a positive whole number.