A sequence is a set of terms, in a definite order, where the terms are obtained by some rule.

A finite sequence ends after a certain number of terms.

An infinite sequence is one that continues indefinitely.

For example: 1, 3, 5, 7, …(This is a sequence of odd

numbers)1st term = 2 x 1 – 1 = 1

2nd term = 2 x 2 – 1 = 33rd term = 2 x 3 – 1 = 5

nth term = 2 x n – 1 = 2n - 1

. .

. .

. .

+ 2

+ 2

NOTATION1st term = u

2nd term =u3rd term =u

nth term =u

. .

. .

. .

1

2

3

n

OR1st term = u

2nd term =u3rd term =u

nth term =u

. .

. .

. .

0

1

2

n-1

FINDINGTHE FORMULA

FORTHE TERMS OFA SEQUENCE

A recurrence relation defines the first term(s) in the sequence and the relation between successive terms.

u = 5u = u +3 = 8u = u +3 = 11

u = u +3 = 3n + 2

.

.

.

1

2

3

n+1

For example:

5, 8, 11, 14, …

1

2

n

What to look forwhen looking for the rule

defining a sequence

Constant difference: coefficient of n is the difference2nd level difference: compare with square numbers(n = 1, 4, 9, 16, …)

3rd level difference: compare with cube numbers(n = 1, 8, 27, 64, …)

None of these helpful: look for powers of numbers(2 = 1, 2, 4, 8, …)Signs alternate: use (-1) and (-1)

-1 when k is odd +1 when k is even

kk

2

3

n - 1

EXAMPLE:

Find the next three terms in the sequence 5, 8, 11, 14, …

EXAMPLE:

The nth term of a sequence is given by x =

a) Find the first four terms of the sequence.

b) Which term in the sequence is ?

c) Express the sequence as a recurrence relation.

1__

2nn

1

1024____

EXAMPLE:

Find the nth term of the sequence +1, -4, +9, -16, +25, …

EXAMPLE:

A sequence is defined by a recurrence relation of the form:M = aM + b.Given that M = 10, M = 20, M = 24, find the value of a and

thevalue of b and hence find M .

n + 1

1 32

4