Sequences, expressions and equations
Continuing number sequences, using letters for unknown numbers, using formulae and solving simple equations.
📌 Key points
- A sequence is a list of numbers that follows a rule. In 5, 8, 11, 14, … the term-to-term rule is "add 3".
- Sequences can go down past zero: 7, 4, 1, −2, −5, … (subtract 3 each time).
- Square numbers (1, 4, 9, 16, 25, …) and triangular numbers (1, 3, 6, 10, 15, …) are special sequences.
- A position-to-term rule gives a term from its position: in 4, 8, 12, 16, … each term is 4 × its position.
- A letter can stand for an unknown number. 3n means 3 × n, and n + 5 means 5 more than n.
- To substitute, put the number in place of the letter: if n = 4, then 3n + 2 = 3 × 4 + 2 = 14.
- A formula is a rule written with letters, e.g. the perimeter of a square is P = 4s.
- Solve an equation with the inverse operation: n + 7 = 15, so n = 15 − 7 = 8.
- An equation is like a balance: whatever you do to one side, do the same to the other side.
📖 Key words
- Sequence
- A list of numbers or shapes that follow a rule.
- Term
- Each number in a sequence.
- Term-to-term rule
- The rule that takes you from one term to the next, e.g. "add 4".
- Position-to-term rule
- The rule that gives a term from its position, e.g. "multiply the position by 4".
- Expression
- Numbers and letters joined by operations, with no equals sign, e.g. 2n + 1.
- Equation
- A statement with an equals sign, where you can find the unknown, e.g. x + 3 = 10.
- Formula
- A rule that shows how to work out one amount from others, e.g. A = l × w.
✏️ Worked example
A row of squares is made from sticks: 1 square uses 4 sticks, 2 squares use 7 sticks and 3 squares use 10 sticks. How many sticks are needed for 6 squares?
- The sequence is 4, 7, 10, … and each term is 3 more than the one before.
- Continue it: 4 squares need 13 sticks, 5 squares need 16 sticks and 6 squares need 19 sticks.
- Check with a rule: sticks = 3 × number of squares + 1, and 3 × 6 + 1 = 19.
- Answer: 19 sticks.
Always test your rule on the terms you are given before you use it to find a new term.
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