Number Sequences and the nth Term

What is a sequence?

A sequence is a list of numbers that follows a rule, such as

3, 7, 11, 15, 19, …

Each number is a term. The first term is 3, the second is 7, and so on. The little dots mean the sequence carries on forever in the same way.

In primary school you learn to spot what comes next by adding on. In algebra we go further: we find a single rule that gives any term instantly. That rule is called the nth term, and finding it uses the substitution ideas from Algebra Basics.

The term-to-term rule and the common difference

The simplest rule is the term-to-term rule — what you add to get from one term to the next. In 3, 7, 11, 15 you add 4 each time. That constant jump is the common difference.

A sequence with a fixed common difference is called a linear (or arithmetic) sequence. You find the common difference by subtracting any term from the one after it:

$$ 7 - 3 = 4, \quad 11 - 7 = 4, \quad 15 - 11 = 4 $$

The difference is 4 every time, confirming it is linear.

Why we want the nth term

The term-to-term rule is slow for big terms. To find the 100th term of 3, 7, 11, … you'd have to add 4 ninety-nine times. The nth term rule fixes this: substitute n = 100 once and you're done.

The nth term is a position-to-term rule. You give it a position n and it returns the term in that position.

Position (n)	1	2	3	4
Term	3	7	11	15

Finding the nth term: the method

For a linear sequence:

1. Find the common difference d. This becomes the coefficient of n, so the rule starts with dn.
2. Compare dn with the actual sequence: work out what dn would give and adjust by adding or subtracting a constant.
3. Write the rule as dn + c.
4. Check by substituting n = 1, 2, 3.

Worked example 1 — an increasing sequence

Find the nth term of 5, 7, 9, 11, …

1. Common difference: 7 − 5 = 2. So the rule starts with 2n.
2. Compare with 2n: the 2-times table is 2, 4, 6, 8. Our sequence is 5, 7, 9, 11 — each term is 3 more than the 2-times table.
3. Write the rule: 2n + 3.
4. Check: n = 1 gives 2(1) + 3 = 5 ✓; n = 2 gives 2(2) + 3 = 7 ✓; n = 3 gives 2(3) + 3 = 9 ✓.

The nth term is 2n + 3.

Worked example 2 — a bigger common difference

Find the nth term of 6, 11, 16, 21, …

1. Common difference: 11 − 6 = 5. Rule starts with 5n.
2. Compare with 5n: the 5-times table is 5, 10, 15, 20. Our sequence is 6, 11, 16, 21 — each term is 1 more.
3. Rule: 5n + 1.
4. Check: n = 1 → 5 + 1 = 6 ✓; n = 2 → 10 + 1 = 11 ✓.

The nth term is 5n + 1.

Worked example 3 — subtracting a constant

Find the nth term of 1, 4, 7, 10, …

1. Common difference: 4 − 1 = 3. Rule starts with 3n.
2. Compare with 3n: the 3-times table is 3, 6, 9, 12. Our sequence is 1, 4, 7, 10 — each term is 2 less.
3. Rule: 3n − 2.
4. Check: n = 1 → 3 − 2 = 1 ✓; n = 3 → 9 − 2 = 7 ✓.

The nth term is 3n − 2.

Worked example 4 — a decreasing sequence

Find the nth term of 20, 17, 14, 11, …

When a sequence goes down, the common difference is negative.

1. Common difference: 17 − 20 = −3. Rule starts with −3n.
2. Compare with −3n: that gives −3, −6, −9, −12. Our sequence is 20, 17, 14, 11 — each term is 23 more than −3n (since 20 − (−3) = 23).
3. Rule: −3n + 23, usually written 23 − 3n.
4. Check: n = 1 → 23 − 3 = 20 ✓; n = 2 → 23 − 6 = 17 ✓.

The nth term is 23 − 3n.

Worked example 5 — using the rule to find a far-off term

The nth term of a sequence is 4n − 1. Find the 25th term.

1. Substitute n = 25 into 4n − 1.
2. 4(25) − 1 = 100 − 1 = 99.

The 25th term is 99 — found in one line, no listing required. This is exactly why the nth term is so useful.

Worked example 6 — is a number in the sequence?

Is 100 a term of the sequence with nth term 3n + 1?

Set the rule equal to 100 and solve for n. If n is a whole number, then 100 is in the sequence.

1. 3n + 1 = 100.
2. Subtract 1: 3n = 99.
3. Divide by 3: n = 33.

Since 33 is a whole number, yes — 100 is the 33rd term. If n had come out as a fraction, the number would not be in the sequence.

Activity — find the nth term of each

1. 4, 6, 8, 10
2. 2, 5, 8, 11
3. 7, 12, 17, 22
4. 0, 4, 8, 12
5. Using nth term 6n − 4, find the 12th term.
6. Is 45 a term of the sequence 5n? Find n if so.

Answers: 1) 2n + 2 2) 3n − 1 3) 5n + 2 4) 4n − 4 5) 6(12) − 4 = 68 6) Yes, 5n = 45 gives n = 9, the 9th term.

Why this matters

The nth term turns a list into a formula, and formulas are how mathematics makes predictions. Once you can write the rule for a sequence, you can find the 1000th term, decide whether a number appears at all, and compare two sequences instantly. It is also your first taste of how patterns become functions, which leads on to functions and graphs.

Keep the method simple: the common difference is the coefficient of n, then adjust with a constant, then check n = 1, 2, 3. Master that and you can describe any linear sequence with a single, powerful rule.