Sequences

Foundation and Higher

The idea

A term-to-term rule gets the next term from the one before. Arithmetic sequences add the same amount each time; geometric sequences multiply by the same amount: $3, 6, 12, 24$ . Know the square, cube, triangular and Fibonacci sequences.

Find the difference between terms: if it is constant, the sequence is arithmetic.
If not, divide each term by the one before: a constant ratio means geometric.
Continue the pattern with the rule you found.

Watch out The term number (position) is not the term itself: the 4th term of $5, 9, 13, \dots$ is $17$ .

See it worked

Worked exampleNon-calculator4 marks

Here are the first four terms of a sequence: $5, 9, 13, 17$ .

(a) Write down the next term of the sequence.

(b) Which is the term-to-term rule?

(c) Is 50 a term in this sequence? You must show how you get your answer.

(a) The terms go up by 4 each time: $17 + 4 = 21$ .
(b) $9 - 5 = 4$ , $13 - 9 = 4$ : add 4 each time.
(c) Every term is 1 more than a multiple of 4: $5 = 4 + 1$ , $9 = 8 + 1$ , and so on.
$50 = 48 + 2$ , so 50 is not in the sequence. (The sequence goes $\ldots, 45, 49, 53$ .)

Answer (a) $21$ ; (b) Add 4; (c) No: the terms are 1 more than a multiple of 4, and 50 is not.

Your turn

Question 1Non-calculator3 marks

Answer each part.

(a) Write down the next square number after 49.

(b) Write down the 5th cube number.

Question 2Non-calculator5 marks

A sequence starts with 3. Each new term is found by multiplying the previous term by 2 and adding 3.

(a) Find the fourth term.

(b) Find the term immediately before the first term if the same rule is used.

Question 3Non-calculator5 marks

Answer each part.

(a) In a sequence, each term after the second is the sum of the two terms before it. The first term is 3 and the second term is $x$ . The fifth term is 30. Work out the value of $x$ .

(b) The first three terms of a geometric sequence are $2, 6, 18$ . Work out the 6th term.

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