Sequences

A sequence is a list of numbers that follows a rule. Each number is a term.

• A term-to-term rule says how to get from one term to the next, such as "add 4".
• A position-to-term rule, or nth term, gives any term from its position n, such as 4n + 3.

To find a term from the nth term, substitute its position for n.

Example: the sequence 7, 11, 15, 19, … has nth term 4n + 3. Find the next two terms and the 10th term.

1. Term-to-term rule: add 4
2. Next two terms: 23, 27
3. 10th term: 4 × 10 + 3 = 43

The nth term is quicker for terms far along: you don't have to work out every term before it.

In a linear sequence, the terms go up (or down) by the same amount each time: the common difference.

1. Find the common difference d. The nth term starts dn.
2. Work out what to add to d × 1 to get the first term.
3. Write dn + that number.

Example: find the nth term of 5, 8, 11, 14, …

1. The difference is 3, so the nth term starts 3n.
2. When n = 1, 3n = 3. The first term is 5, so add 2.
3. 3n + 2

Example: find the nth term of 20, 17, 14, 11, …

1. The difference is −3, so it starts −3n.
2. When n = 1, −3n = −3. To get 20, add 23.
3. 23 − 3n

Check the second term: 23 − 3 × 2 = 17.

Set the nth term equal to the number and solve. If n is a whole number, the number is in the sequence, and n is its position.

1. Write nth term = the number.
2. Solve for n.
3. Whole number: yes. Not a whole number: no.

Example: the nth term of a sequence is 3n + 2. Are 100 and 131 in the sequence?

1. 3n + 2 = 100, so 3n = 98 and n = 32.67…
2. n isn't a whole number, so 100 is not in the sequence
3. 3n + 2 = 131, so 3n = 129 and n = 43
4. 131 is the 43rd term

Learn to recognise these:

In a geometric sequence, the number you multiply by is the common ratio.

Example: find the next two terms of 2, 5, 7, 12, … and of 81, 27, 9, …

1. 2 + 5 = 7 and 5 + 7 = 12: Fibonacci-type, so 19, 31
2. 27 ÷ 81 = ⅓: geometric with ratio ⅓, so 3, 1

The common ratio can be a surd. Find it by dividing a term by the one before, then simplify.

1. Divide the second term by the first to get the ratio r.
2. Multiply by r to get each new term, using √a × √a = a.

Example: a geometric sequence starts 5, 5√2, 10, … Find the common ratio and the next two terms.

1. r = 5√2 ÷ 5 = √2
2. Check: 5√2 × √2 = 5 × 2 = 10
3. 10 × √2 = 10√2
4. 10√2 × √2 = 10 × 2 = 20
5. Ratio √2; next two terms 10√2, 20

Every second term is a whole number, because multiplying by √2 twice is multiplying by 2.

In a quadratic sequence, the differences change, but the second differences (the differences of the differences) are all the same. The nth term has an n² term.

1. Find the second difference. The nth term starts (second difference ÷ 2)n².
2. Subtract that n² part from each term.
3. Find the nth term of what's left (a linear sequence) and add it on.

Example: find the nth term of 4, 9, 18, 31, 48, …

1. Differences: 5, 9, 13, 17. Second differences: 4. So it starts 2n².
2. 2n²: 2, 8, 18, 32, 50
3. Sequence − 2n²: 2, 1, 0, −1, −2, which is −n + 3
4. 2n² − n + 3

Check the 3rd term: 2 × 9 − 3 + 3 = 18.