The nth Term Made Simple (11+)

A9The nth Term

Jump straight to the 100th term.

The nth term is a formula for ANY term — no counting up needed. (11+ stretch)

3, 7, 11, 15, … what is the 100th term?

gaps of 4 → the formula starts 4n

4 × 1 = 4, but the 1st term is 3 → subtract 1

nth term = 4n − 1 → 100th = 4 × 100 − 1 = 399

nth term = (gap) × n ± adjustment. Test it on the first two terms.

Learn the 3-Step Method

1The gap gives the times table.

Gaps of 4 → the sequence shadows the 4 times table: 4n.

gap 4 → 4n

2Compare with the real terms.

4n gives 4, 8, 12… but our terms are 3, 7, 11 — each is 1 LESS.

1 less → −1

3Write and test the formula.

4n − 1. Test: n = 1 → 3 ✓, n = 2 → 7 ✓. Now jump anywhere: n = 100 → 399.

4n − 1

💡 Key IdeaThe power of nth term: the 100th term without 99 additions. That's why the 11+ loves it.

Worked Example

Find the nth term of 5, 8, 11, 14 — and its 50th term.

Gap

gaps of 3 → 3n

→

Adjust

3n gives 3, 6, 9…
terms are 2 more → 3n + 2

→

✓ 50th term

3 × 50 + 2 = 152

152

Watch Out! Common Mistakes

Using the first term as the multiplier.

3, 7, 11 → 3n ✕

It’s the GAP! Gaps of 4 → 4n (then adjust): 4n − 1.

Not testing the formula.

Test n = 1 and n = 2! A formula that fails the first two terms is wrong — the test takes seconds and catches everything.

Adjusting the wrong way.

terms 1 less than 4n → 4n + 1 ✕

Direction! Terms BELOW the times table need MINUS: 4n − 1.

⬇ Cheat Sheet

THE NTH TERM

Gap = the times table: gap 4 → 4n.
Compare 4n's values with the real terms.
Adjust: terms above → +, below → −.
Test on n = 1 and n = 2 before using.

Examples: 3,7,11 → 4n − 1 (100th = 399) · 5,8,11 → 3n + 2 (50th = 152).

📖 Continue Your Journey

✓ A1 What Is Algebra?✓ A2 Function Machines ✓ A3 Writing Expressions ✓ A4 Substitution ✓ A5 Simplifying Expressions ✓ A6 One-Step Equations ✓ A7 Two-Step Equations ✓ A8 Sequences A9 The nth Term A10 Formulae A11 Bar Models for Unknowns A12 Algebra Problem Solving

→

Next Lesson: A10 FormulaeRules you can calculate with.

📄 Free printable: Download the The nth Term cheat sheet (PDF) — keep it by your child's desk.

Quick summary

The gap between terms gives the times table: gaps of 4 → 4n.
Compare the times table with the actual terms and adjust: 4n − 1.
Test the formula on the first two terms before trusting it.
Then jump anywhere: the 100th term of 4n − 1 is 399.

Frequently asked questions

What is the nth term of a sequence?

A formula giving any term directly from its position: for 3, 7, 11, 15 the nth term is 4n − 1, so the 100th term is 4 × 100 − 1 = 399.

How do you find the nth term?

Use the gap as the multiplier (gaps of 4 → 4n), compare with the actual terms to find the adjustment (−1 here), then test on the first two terms.

Is the nth term on the KS2 curriculum?

It's beyond the standard Year 6 objectives but appears in grammar-school 11+ papers, so we teach it as a clearly-marked stretch topic.