Algebra · KS3

Linear Sequences

5, 8, 11, 14, … What's the 100th number? You could add 3 ninety-nine times, or you could write one formula, the nth term, that jumps straight to any position. Finding it takes two numbers: how much you add each time, and where you'd have started.

National curriculum KS3: generate terms of a sequence from a term-to-term or position-to-term rule; recognise arithmetic sequences and find the nth term

1 · Patterns that grow by the same amount

A sequence is a list of numbers in order; each one is a term, and its place in the list is its position, n. A linear (or arithmetic) sequence goes up or down by the same amount every time. That amount is the common difference, d.

Here it is as a growing pattern. Each new shape adds the same number of sticks. Slide to build more.

2 · Term-to-term versus position-to-term

There are two ways to describe the same sequence. The term-to-term rule says how to get the next term from the one before: "start at 5, add 3". Easy to state, useless for the 100th term. The position-to-term rule (the nth term) says how to get any term straight from its position: "3n + 2". Harder to find, but then it does everything.

3 · Finding the nth term

Given the sequence, how do you find the formula? Two numbers.

  1. The common difference is the coefficient of n. If the sequence goes up by 3, the formula starts 3n. (Think: the 3 times table goes up by 3, and so does anything built on it.)
  2. The adjustment: compare the sequence with the 3 times table. 3, 6, 9, 12 versus 5, 8, 11, 14. Every term is 2 more. So the formula is 3n + 2.

A faster way to get the adjustment: step backwards one place from the first term to find the zeroth term. 5 − 3 = 2. That's the "+ 2". The zeroth term is where the sequence would have started if there were a position 0, and it's always the constant in the formula.

The classic mistake. Seeing 5, 8, 11, 14 and writing n + 3 because "you add 3". The term-to-term rule is add 3; the nth term needs 3n, because by position n you have added 3 a total of (n − 1) times. Check any formula by substituting n = 1 and n = 2: n + 3 gives 4, 5. Wrong. 3n + 2 gives 5, 8. Right.

4 · It's a straight line in disguise

Plot the terms against their positions and they sit on a straight line. The common difference is the gradient and the zeroth term is the y-intercept. The nth term dn + c is y = mx + c with x restricted to 1, 2, 3, … That's why it's called a linear sequence.

5 · Is a number in the sequence?

Once you have the nth term, you can answer "is 302 in the sequence 5, 8, 11, …?" without listing a hundred terms. Set the formula equal to the number and solve for n (see the linear equations page). If n is a positive whole number, yes, and you know which position. If it's a fraction or negative, no.

6 · Practice

A random linear sequence; type its nth term as coefficient and constant. Working appears after you check.

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7 · Where sequences show up

Saving and paying

£20 in the account, £5 added weekly: 25, 30, 35, … The nth term 5n + 20 tells you the balance after n weeks without a spreadsheet. (Compound interest is a geometric sequence: multiply, not add.)

Seating and stacking

Tables pushed together in a row seat 4, 6, 8, 10 people: 2n + 2. Chairs around hexagonal tables, fence posts and panels, tiles around a pool: all pattern questions, all linear.

Programming

for i in range(5, 100, 3) generates 5, 8, 11, … The nth term is how you compute the i-th element directly: 5 + 3*i. Array indexing in memory is a linear sequence of addresses.

Timetables

A bus every 12 minutes from 7:04: the nth bus leaves at 12n − 8 minutes past 7. Which bus is at 9:16? Solve 12n − 8 = 136: n = 12.

Music and poetry

Bar numbers, syllable counts in a haiku, the fret positions on a guitar (geometric again). Pattern-spotting is the same skill.

GCSE and beyond

Quadratic sequences (second differences constant) at GCSE Higher; arithmetic series, the sum of n terms, at A-level: Sₙ = n/2 (2a + (n − 1)d) is this page scaled up.

Worked examples

Find the nth term of 7, 11, 15, 19
Common difference: +4 each time, so the formula starts 4n.
Zeroth term: 7 − 4 = 3.
nth term = 4n + 3. Check: n = 1 → 7 ✓, n = 2 → 11 ✓
Find the nth term of 20, 17, 14, 11
Common difference: −3 (going down), so −3n.
Zeroth term: 20 + 3 = 23 (step backwards means adding 3 when d is negative).
nth term = 23 − 3n (same as −3n + 23). Check: n = 1 → 20 ✓, n = 4 → 11 ✓
Is 150 in the sequence 4n + 2?
Set equal: 4n + 2 = 150.
Solve: 4n = 148, n = 37.
37 is a positive whole number, so yes: 150 is the 37th term.
Exam-style: the 3rd term of a linear sequence is 14 and the 7th term is 30. Find the nth term.
From the 3rd to the 7th is 4 steps and the terms rise by 30 − 14 = 16. So d = 16 ÷ 4 = 4.
Step back from the 3rd term to the zeroth: 14 − 3 × 4 = 2.
nth term = 4n + 2. Check: n = 3 → 14 ✓, n = 7 → 30 ✓

Quick check

What you've learned

A linear sequence changes by the same amount each step. A term-to-term rule describes that step; the nth term is a formula dn + c that jumps to any position. d is the common difference and becomes the coefficient of n; c is the zeroth term, found by stepping back once from the first term. Check by substituting n = 1 and 2. Plotted against n, the terms lie on the line y = dx + c, which is why they're called linear. Setting the formula equal to a number and solving for n tells you whether, and where, that number appears.