Geometric Sequences

A geometric sequence is a sequence where each # in the seq. is multiplied by a constant (which is called the common ratio (r))

To find r, divide any term in the sequence by the previous term

Geometric Sequences

General Formula:1

1 n

n raa

Geometric Sequences

Find the 11th term of the geo. Sequence listed below

64, -32, 16, -8,…a11

11

nn raa

Geometric Sequences

Find the 6th term of the geo. sequence listed below

3, -15, 75, ..a6

11

nn raa

Geometric Sequences

Write an equation for the nth term

3, 12, 48, 192...an

11

nn raa

Geometric Sequences

Find the 10th term of the sequence if

a4=108

r=3

11

nn raa

Geometric Sequences

Find the 7th term of the sequence if

a3=96

r=2

11

nn raa

Geometric Means

Geometric means are the missing terms between two non-successive terms in a geo. Sequence

Find 3 geometric means between 2.25 and 576

Geometric Means

Find 5 geometric means between ½ and 1/1458

Geometric Series

A series that is associated with a geometric sequence

r

raS

n

n

1

11

Geometric Series

Find the sum of the first 6 terms of the geometric series

3 + 6 + 12 + 24 +…

r

raS

n

n

1

11

Geometric Series

Find the first term of the series if the S8=39,360 and r=3

r

raS

n

n

1

11

Geometric Series

Find the sum of the first 8 terms of

1+x+x2+x3+…

r

raS

n

n

1

11

Sigma Notation

More concise (less time consuming) notation for writing out a series

n) case, (in this variable theissummation ofindex the

10" to1 from goesn as3n of sum the"

3

30...12963:10

1

n

n

ex

Sigma Notation

8

5

43j

j

Sigma Notation

10

3

12k

k

Sigma Notation

5

1

1

2

1

n

n

Write in sigma notation

1 + 3 + 5 + 7

Write in sigma notation

2 + 4 + 6 + 8 + 10

Write in sigma notation

3 + 6 + 12 + 24 + 48

Write in sigma notation

-3 + 9 + -27 + 81 + -243

Write in sigma notation

-2 + 4 + -8 + . . . +256

Infinite Geometric Series

In an infinite series, Sn approaches some limit as n becomes very large. That limit is defined to be the sum of the series. If an infinite series has a sum, it is said to converge.

A series converges (or has a sum) if and only if lrl < 1

Does the geom. series have a sum?

...81

1

27

1

9

1

3

11

Does the geom. series have a sum?

...625

3

125

3

25

3

5

33

Does the sum of each term approach some limit?

12+

23+

89+

3227

+. . . .

To find the sum of an infinite series

Make sure a limit exists first

r

aSn

1

1

An infinite series in sigma notation—find the sum

1

1 5

124

n

n

An infinite series in sigma notation—find the sum

1

1 2

15

n

n

Writing a repeating decimal as a fraction

39.0

Writing a repeating decimal as a fraction

127.0

A different kind of sequence. . .

(𝑥+𝑦 )𝑛

Expand

¿

Pascal’s Triangle

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1First row is used for anything to the zero

power.Used for the coefficients of each term of the

expanded binomial

Expand using Pascal’s Triangle

52 yx

Expand using Pascal’s Triangle

423 yx