unit 7: sequences and series. sequences a sequence is a list of #s in a particular order if the...

Unit 7: Sequences and Series

Sequences

A sequence is a list of #s in a particular orderIf the sequence of numbers does not end,

then it is called an infinite sequenceEach # in a sequence is called a termEx. 3, 5, 7, 9….a1=3a2=5

Arithmetic Sequences

A sequence in which each term after the first term is found by adding a constant (called the common difference (d)), to the previous term

Ex. Find the common difference (one term minus the previous term)

55, 49, 43, 37, 31, 25,19

Arithmetic Sequences

Formula for the general pattern for any arithmetic sequence:

differencecommond

termsofn

termtheofvaluea

termntheofvaluea

where

dnaa

st

thn

n

#

1

...

)1(

1

1

Arithmetic Sequences

Write a equation for the nth term of the following sequence

...,35,26,17,8

dnaan )1(1

Write a equation for the nth term of the following sequence

...,2

7,3,

2

5,2

Arithmetic Sequences dnaan )1(1

Arithmetic Sequences

Find a15

,...3,5,7,9

dnaan )1(1

Arithmetic Sequences

In the sequence below, which term has a value of 286?

...,14,10,6,2

dnaan )1(1

Arithmetic Sequences

What is the value of the first term if the 9th and 10th terms are 4 and 2 consecutively?

dnaan )1(1

Arithmetic Sequences

If the 3rd term of an arithmetic sequence is 8 and the 16th term is 47, find a1 and d

dnaan )1(1

Arithmetic Means

Terms between any two non-successive terms of an arithmetic sequence

Find 4 arithmetic means between 16 and 91

dnaan )1(1

Arithmetic Means

Find 1 arithmetic mean between 50 and -120

dnaan )1(1

Results-4A

0123

• Concerns: arithmetic means, notation, 2x problem, more practice

Arithmetic Series

A series is the indicated sum of the terms of a sequence

If the seq. is 18, 22, 26, 30The arith series is 18 + 22 + 26 + 30

Arithmetic Series

To find the sum of an arithmetic series,

nn aan

S 12

Arithmetic Series

Find the sum of the first 100 positive integers.

nn aan

S 12

Arithmetic Series

Find the sum of the first 20 even numbers beginning with 2.

nn aan

S 12

Arithmetic Series

Find the sum of 34+30+26+…+2 nn aa

nS 12

Arithmetic Series

Find the sum of 6+13+20+27…+97 nn aa

nS 12

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

What is r if the sum of the first 6 terms in a geo series is 11,718 and the first term is 3

Hint: solve by doing an intersection on the graph

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

Geometric Series

Find the a1 if Sn=165, an=48, r=-2/3

Hint: an=a1.rn-1

So, r.an=a1.rn-1.ran.r=a1.rn (now

substitute into the sum formula)

r

raS

n

n

1

11

r

raaS nn

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 + 16 + -32 + 64

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?

...64

1

32

1

16

1

8

1

4

1

2

1

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

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 1*First 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

Writing a repeating decimal as a fraction

39.0

Writing a repeating decimal as a fraction

127.0