GCSE · Algebra

Sequences

A sequence is a list of numbers that follows a rule, like 3, 7, 11, 15, … Each number is a term. Some sequences add the same amount each time, some multiply, some add the previous two terms together, and some grow by a changing amount. Find the rule and you can predict any term, even the millionth.

GCSE subject content: generate terms from term-to-term and position-to-term rules; recognise triangular, square and cube numbers, arithmetic progressions, Fibonacci-type sequences, quadratic sequences and simple geometric progressions; find the nth term of linear sequences; Higher: geometric progressions with a surd ratio; the nth term of quadratic sequences

Recap from KS3. In a linear (arithmetic) sequence the gap between terms is always the same. Its nth term is dn + c, where d is that gap. If that's new to you, start with Linear sequences. Here we recap it quickly and then go further.

1 · Two kinds of rule

A term-to-term rule tells you how to get from one term to the next: "start at 3 and add 4". A position-to-term rule (the nth term) tells you how to get any term straight from its position n: "4n − 1". The first is easy to continue; the second lets you jump straight to the 100th term. Here are both rules for the same sequence, side by side.

start at then show term number n =

2 · Linear sequences and the nth term

For a linear sequence, the common difference d (the amount added each time) is the number in front of n. Why? Because "dn" goes up by d every step, just like the sequence. Then compare the sequence with the d times table to find the adjustment c. For 3, 7, 11, 15: d = 4; the 4 times table is 4, 8, 12, 16; the sequence is 1 less every time, so the nth term is 4n − 1.

Find the nth term of:

nth term = n +

Common mistakes. Using the first term as c: for 3, 7, 11, … the nth term is not 4n + 3 (that gives 7 when n = 1). Using the difference with the wrong sign: for 20, 17, 14, … d = −3, so it's −3n + 23 (or 23 − 3n). To test whether a number is in the sequence, set the nth term equal to it and solve: if n is a whole number, it's in.

3 · Geometric sequences

In a geometric sequence you multiply by the same number each time. That number is the common ratio r, found by dividing any term by the one before: r = term ÷ previous term. If r > 1 the terms grow (fast!); if r is a fraction between 0 and 1 they shrink towards zero. (Higher) r can even be a surd like √2: then every other term is a whole number.

first term ratio r =

ratio r = next term =

4 · Fibonacci-type sequences

In a Fibonacci-type sequence each term is the sum of the two before it. The famous Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, … but you can start with any two numbers. With letters, if the first two terms are a and b, the sequence is a, b, a + b, a + 2b, 2a + 3b, … Exam questions give you two of the later terms and ask you to work back to a and b with simultaneous equations.

a = b =

5 · Triangular, square and cube numbers

Some sequences are named after the shapes they make. Square numbers 1, 4, 9, 16, … are n². Cube numbers 1, 8, 27, 64, … are n³. Triangular numbers 1, 3, 6, 10, 15, … are the number of dots in a triangle with n rows: add 1 + 2 + 3 + … + n. Two copies of a triangle fit together into an n by (n + 1) rectangle, so the nth triangular number is n(n + 1)/2.

n = show two triangles make a rectangle

6 · Quadratic sequences and their nth term Higher

In 4, 9, 18, 31, 48, … the gaps (5, 9, 13, 17) are not constant, but the gaps between the gaps are: 4, 4, 4. When the second differences are constant, the sequence is quadratic: its nth term is an² + bn + c. Why? For n², the terms 1, 4, 9, 16, 25 have first differences 3, 5, 7, 9 and second differences 2: so an² always has second difference 2a.

Method: (1) second difference ÷ 2 = a; (2) subtract an² from each term; (3) what's left is linear: find its nth term bn + c; (4) add them.

Sequence

Your turn: find the nth term of

n² + n +

Worked examples

Find the nth term of 5, 8, 11, 14, … and the 100th term

Common difference: +3, so the nth term starts 3n.

Compare with 3n: 3, 6, 9, 12. The sequence is 2 more each time.

nth term: 3n + 2. Check n = 1: 5 ✓.

100th term: 3 × 100 + 2 = 302.

Is 100 a term of the sequence 3n + 2?

Set equal: 3n + 2 = 100, so 3n = 98, n = 32.67…

Not a whole number, so 100 is not in the sequence. (The 32nd term is 98 and the 33rd is 101.)

48, 24, 12, 6, … Find the next two terms.

Ratio: 24 ÷ 48 = ½ (and 12 ÷ 24 = ½ ✓). It's geometric with r = ½.

Next terms: 6 × ½ = 3, then 3 × ½ = 1.5.

2, 2√3, 6, 6√3, … Find the next term. Higher

Ratio: 2√3 ÷ 2 = √3.

Next term: 6√3 × √3 = 6 × 3 = 18 (because √3 × √3 = 3).

A Fibonacci-type sequence has 3rd term 7 and 5th term 19. Find the first two terms.

Letters: a, b, a + b, a + 2b, 2a + 3b.

Equations: a + b = 7 and 2a + 3b = 19.

Solve: double the first: 2a + 2b = 14. Subtract from the second: b = 5. Then a = 2.

Check: 2, 5, 7, 12, 19 ✓

Find the nth term of 4, 9, 18, 31, 48, … Higher

First differences: 5, 9, 13, 17. Second differences: 4, 4, 4: quadratic.

a = 4 ÷ 2 = 2, so the nth term starts 2n². 2n² gives 2, 8, 18, 32, 50.

Subtract: 4 − 2 = 2, 9 − 8 = 1, 18 − 18 = 0, 31 − 32 = −1, 48 − 50 = −2. This linear sequence is −n + 3.

nth term: 2n² − n + 3. Check n = 4: 32 − 4 + 3 = 31 ✓

What is the 10th triangular number?

Formula: n(n + 1)/2 = 10 × 11 ÷ 2 = 55.

Check by adding: 1 + 2 + … + 10 = 55 ✓ (pair 1 + 10, 2 + 9, … five pairs of 11).

Quick check

What you've learned

A term-to-term rule gets the next term from the previous one; a position-to-term rule (nth term) gets any term from its position.
Linear: constant difference d; nth term dn + c. To test whether a number is a term, solve nth term = number and see if n is a whole number.
Geometric: constant ratio r = term ÷ previous term; it can be a fraction, or (Higher) a surd.
Fibonacci-type: each term is the sum of the previous two: a, b, a + b, a + 2b, 2a + 3b, …
Square n², cube n³, triangular n(n + 1)/2.
Higher Quadratic: constant second difference 2a. Find a, subtract an², then find the linear nth term of what's left.