Math Playground

Algebra › Pattern matching

What comes next?

The first step into algebra is noticing patterns. Numbers, colors, shapes — spot the rule, predict the next item.

Why patterns?

Algebra is the maths of rules. Before you can write a rule with letters, you have to see the rule in the numbers. Patterns are how that skill grows: spot the regularity, predict the next item, write down what's going on.

Three kinds of pattern

  • Repeating — the same chunk repeats: 🔴 🔵 🟢 🔴 🔵 🟢 …
  • Growing — each item is bigger than the last by some fixed rule: 2, 4, 6, 8, 10 (add 2) · 1, 3, 9, 27, 81 (multiply by 3).
  • Position-based — the value depends on its position: 1, 4, 9, 16, 25 (square the position) · 1, 1, 2, 3, 5, 8, 13 (Fibonacci — sum the previous two).

Worked examples

Add the same number each time

3, 7, 11, 15, ? Each term is +4. So the next term is 19. The rule in words: start at 3 and add 4 to get the next.

Multiply by the same number each time

2, 6, 18, 54, ? Each term is × 3. So next is 162. Adding always gives arithmeticpatterns; multiplying gives geometric ones.

The picture grows by a known shape

A staircase of squares: row 1 has 1 square, row 2 has 3 squares, row 3 has 5, row 4 has 7. The differences are 2, 2, 2 — so it's an arithmetic pattern. Row 10 will have 1 + 9 × 2 = 19 squares.

From pattern to formula

Once you spot the rule, you can write it with letters. The "add 4" pattern starting at 3 gives the formula 3 + 4(n − 1) for term n. That's algebra in disguise — the letters do the counting for you so you can jump straight to term 100 without writing all 99 in between.

Spotting traps

  • Two-step patterns. 1, 2, 4, 7, 11, 16 — the differences are 1, 2, 3, 4, 5 (an arithmetic pattern of the differences). The pattern is "add the next counting number."
  • The same pattern, different starting point. "Every third term" is one rule. Look at all the terms before committing.
  • Many rules fit the same start. 1, 2, 4 could be ×2 (so 8 next), or +1 then +2 then +3 (so 7 next). You usually need 4+ terms to be confident.

Try it

  1. What's next? 5, 10, 20, 40, ?
  2. What's next? 1, 4, 9, 16, ?
  3. What's next? 1, 1, 2, 3, 5, 8, ?
  4. What's the rule for 11, 14, 17, 20?

Answers: 1) 80 (× 2). 2) 25 (the squares). 3) 13 (Fibonacci). 4) Start at 11, add 3 each time.