SEQUENCES

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# SEQUENCES - PowerPoint PPT Presentation

SEQUENCES. 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. 1, 3, 5, 7, … (This is a sequence of odd numbers).

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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.

1, 3, 5, 7, …

(This is a sequence of odd numbers)

1st term = 2 x 1 – 1 = 1

For example:

+ 2

2nd term = 2 x 2 – 1 = 3

+ 2

3rd term = 2 x 3 – 1 = 5

. .

. .

. .

nth term = 2 x n – 1 = 2n - 1

NOTATION

1

2nd term = u

1st term = u

2

3rd term = u

3

. .

. .

. .

nth term = u

n

OR

0

2nd term = u

1st term = u

1

3rd term = u

2

. .

. .

. .

nth term = u

n-1

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

For example:

5, 8, 11, 14, …

1

u = u +3 = 8

u = 5

1

2

u = u +3 = 11

3

2

.

.

.

u = u +3 = 3n + 2

n+1

n

What to look for

when looking for the rule

defining a sequence

2

• Constant difference: coefficient of n is the difference
• 2nd 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

3

n - 1

k

k

EXAMPLE:

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

1

__

n

n

2

EXAMPLE:

The nth term of a sequence is given by x =

• Find the first four terms of the sequence.

b) Which term in the sequence is ?

c) Express the sequence as a recurrence relation.

1

____

1024

EXAMPLE:

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

n + 1

1

2

3

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 the

value of b and hence find M .

4

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