SEQUENCESOBJECTIVES:• Write the first several terms of a sequence• Write the terms of a sequence defined by a

Recursive Formula• Use Summation Notation• Find the Sum of a Sequence

SEQUENCES Infinite sequence: is a function whose

domain is the set of positive integers. The function values, or terms, of the sequence are represented by

Finite sequence: sequences whose domains consist only of the first positive integers

The letter with a subscript is used to represent function values of a sequence, rather than the usual function notation. The notation represents the nth term, or general term, of a sequence.

na

1 2 3, , , , na a a a

n

na

EX 1: WRITE THE FIRST FIVE TERMS OF A SEQUENCE FROM THE GENERAL TERM A)

B)

2 12nn

13 1

n

n na

EX 2: WRITE DOWN THE NTH TERM OF EACH SEQUENCE A)

B)

1 2 3 4, , , ,2 3 4 5

1, 2,3, 4,5, 6,

RECURSIVE FORMULA Recursive Formula: defines the nth term of a

sequence as a function of the previous term or terms and specifies the nth term by a formula or equation that involves one or more of the terms preceding it.

is the first term is the previous term

1a

1na

EX 3: WRITE THE FIRST FIVE TERMS OF A SEQUENCE DEFINED RECURSIVELY A)

B)

C)1

1

53 2n n

aa a

1

2

1 2

12

n n n

aa

a a a

21 !

n

na n

SUMMATION NOTATION Summation Notation: the sum of the first

terms of a sequence is represented by the summation notation

where is the index of summation, is the upper limit of summation, and 1 is the lower limit of summation.

Expanding the summation notation is when we write out a sum that is given in summation notation.

1 2 3 41

n

i ni

a a a a a a

ni

EX 4: WRITE OUT EACH SUM A)

B)

6

2

1

1i

i

2

1 2

n

k

k

EX 5: EXPRESS EACH SUM USING SUMMATION NOTATION A)

B)

1 2 3 132 3 4 13 1

1 1 113 9 27