USING AND WRITING SEQUENCES

The numbers (outputs) of a sequence are called terms.

You can think of a sequencesequence as a set of numbers writtenin a specific order. (Any sequence can be defined as afunction whose domain is the set of natural numbers.)

The domain gives the relative position of each term.

1 2 3 4 5 DOMAIN:

3 6 9 12 15RANGE:The range gives the terms of the sequence.

This is a finite sequence having the rule

an = 3n,where an represents the nth term of the sequence.

USING AND WRITING SEQUENCES

n

an

Writing Terms of Sequences

Write the first six terms of the sequence an = 2n + 3.

SOLUTION

a 1 = 2(1) + 3 = 5 1st term

2nd term

3rd term

4th term

6th term

a 2 = 2(2) + 3 = 7

a 3 = 2(3) + 3 = 9

a 4 = 2(4) + 3 = 11

a 5 = 2(5) + 3 = 13

a 6 = 2(6) + 3 = 15

5th term

Writing Terms of Sequences

Write the first six terms of the sequence f (n) = (–2)

n – 1 .

SOLUTION

f (1) = (–2) 1 – 1 = 1 1st term

2nd term

3rd term

4th term

6th term

f (2) = (–2) 2 – 1 = –2

f (3) = (–2) 3 – 1 = 4

f (4) = (–2) 4 – 1 = – 8

f (5) = (–2) 5 – 1 = 16

f (6) = (–2) 6 – 1 = – 32

5th term

3 + 6 + 9 + 12 + 15 = ∑ 3i5

i = 1

SUMMATIONSUMMATION Notation (aka SIGMASIGMA Notation)

5

i = 1∑3i

Is read as “the sum of 3i from i equals 1 to 5.”

index of summation lower limit of summation

upper limit of summation

SUMMATIONSUMMATION Notation (aka SIGMASIGMA Notation)

The index of summation does not have to be ii. Any letter can be used. Also, the index does not have to begin at 1 (but often does).

Writing Series with Summation Notation

Write this series using summation notation:

5 + 10 + 15 + + 100. . .

SOLUTION

Notice that the first term is 5 (1), the second is 5 (2),the third is 5 (3), and the last is 5 (20). So the termsof the series can be written as:

ai = 5i where i = 1, 2, 3, . . . , 20

The summation notation is: 20

1

5i

i

Example: Write the series represented by the summation notation . Then find the sum.

3

0

12

!i i

SOLUTION:

12 12 12 12

0 1 2 3! ! ! !

12 12 12 12

1 1 2 6

32