Sequences

Sequence

There are 2 types of Sequences

Arithmetic:You add a common difference each time.

Geometric:You multiply a common ratio each time.

Arithmetic SequencesExample:• {2, 5, 8, 11, 14, ...}

Add 3 each time• {0, 4, 8, 12, 16, ...}

Add 4 each time• {2, -1, -4, -7, -10, ...}

Add –3 each time

Arithmetic Sequences• Find the 7th term of the sequence:

2,5,8,…Determine the pattern:Add 3 (known as the common difference)Write the new sequence:2,5,8,11,14,17,20

So the 7th number is 20

Arithmetic Sequences

• When you want to find a large sequence, this process is long and there is great room for error.

• To find the 20th, 45th, etc. term use the following formula:

an = a1  + (n - 1)d

Arithmetic Sequences

an = a1  + (n - 1)dWhere:

a1 is the first number in the sequence n is the number of the term you are

looking for d is the common difference

an is the value of the term you are looking for

Arithmetic Sequences• Find the 15th term of the sequence:

34, 23, 12,…

Using the formula an = a1  + (n - 1)d,

a1 = 34d = -11n = 15

an = 34 + (n-1)(-11) = -11n + 45

a15 = -11(15) + 45

a15 = -120

Arithmetic Sequences

Melanie is starting to train for a swim meet. She begins by swimming 5 laps per day for a week. Each week she plans to increase her number of daily laps by 2. How many laps per day will she swim during the 15th week of training?

Arithmetic Sequences• What do you know?

an = a1  + (n - 1)d

a1 = 5

d= 2

n= 15

t15 = ?

Arithmetic Sequences

• tn = t1  + (n - 1)d

• tn = 5  + (n - 1)2

• tn = 2n  + 3

• t15 = 2(15) + 3

• t15 = 33

During the 15th week she will swim

33 laps per day.

Geometric Sequences

• In geometric sequences, you multiply by a common ratio each time.

• 1, 2, 4, 8, 16, ... multiply by 2• 27, 9, 3, 1, 1/3, ...

Divide by 3 which means multiply by 1/3

Geometric Sequences• Find the 8th term of the sequence:

2,6,18,…Determine the pattern:Multiply by 3 (known as the common

ratio)Write the new sequence:2,6,18,54,162,486,1458,4374So the 8th term is 4374.

Geometric Sequences

• Again, use a formula to find large numbers.

•         an = a1 • (r)n-1

Geometric Sequences

• Find the 10th term of the sequence :

4,8,16,…

an = a1 • (r)n-1

• a1 = 4• r = 2• n = 10

Geometric Sequences

an = a1 • (r)n-1

a10 = 4 • (2)10-1

a10 = 4 • (2)9

a10 = 4 • 512

a10 = 2048

Geometric Sequences

• Find the ninth term of a sequence if a3 = 63 and r = -3a1= ?n= 9r = -3a9 = ?There are 2 unknowns so you must…

Geometric Sequences

• First find t1. • Use the sequences formula substituting t3 in

for tn.  a3 = 63• a3 = a1 • (-3)3-1 • 63 = a1 • (-3)2

• 63= a1 • 9• 7 = a1

Geometric Sequences

• Now that you know t1, substitute again to find tn.

an = a1 • (r)n-1

a9 = 7 • (-3)9-1

a9 = 7 • (-3)8

a9 = 7 • 6561

a9 = 45927

SequenceA sequence is a set of numbers

in a specific order

Infinite sequence

Finite sequence

,...,...,,,, 4321 naaaaa

naaaaa ,...,,,, 4321

Sequences – sets of numbers

Notation:

represents the formula for finding terms

term numberna

n

Examples:

If 2 3, find the first 5 terms.na n

term. 20th the find ,13 If nan

th4

nd32

is the notation for the 4 term

is the notation for the 32 term

a

a

Ex.1 Find the first four terms of the sequence

23 nan

12)1(31 a42 a

73 a

104 a

First term

Second term

Third term

Fourth term

Ex. 2

12

)1(

n

an

n

Find the first four terms of the sequence

Writing Rules for Sequences

We can calculate as many terms as we want as long as we know the rule or equation for an.

Example:

3, 5, 7, 9, ___ , ___,……. _____ .an = 2n + 1

Writing Rules for Sequences

Try these!!!

3, 6, 9, 12, ___ , ___,……. _____ .

1/1, 1/3, 1/5, 1/7, ___ , ___,……. _____ .

an = 3n, an = 1/(2n-1)