sequences series and binomial theorem

8/12/2019 Sequences Series and Binomial Theorem

1/7

Sequences, Series and Binomial theorem

A sequenceis a set of numbers arranged in a definite order.

1,2, 3, , , where 1, 2, 3, , , are called terms.

Arithmetic Sequence and Series

A sequence {} in which each term differs from the previous one by the sameconstant (common difference) is called arithmetic sequence.

+ =The general termcan be found using the formula

= + ( )

The Sum () of the first terms of an arithmetic sequence {} is calledarithmetic seriesand given by = ( + )or = [ + ( )]

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Example

An arithmetic sequence has a 1stterm of 80 and a 24thterm of 172. Find the 74thterm,

an expression for the general term and the sum .

Solution

Using the formula = + ( )we have that

24 =1 + (24 1) =172 80

23=

92

23= 4

Thus,74 =1 + (74 1) = 8 0 + (74 1)4 = 372

The general term is = + ( )and the sum = [ + ( )]





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Geometric Sequence and Series

A sequence {} in which each term can be obtained from the previous one bymultiplying by a non-zero constant (common ratio) is called geometricsequence.

+ =The general termcan be found using the formula

=

The Sum () of the first n terms of an geometric sequence {} is calledgeometric seriesand given by

=()

, || < 1 or =()

, || > 1

The Sum to infinity () of geometric series is given by = , || < 1





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4/7

Example

A geometric sequence has a 4thterm of 32 and a 7thterm of 256. Find the 10thterm, an

expression for the general term and the sum .

Solution

Using the formula =we have that4 =141 32 =13 (1)

7 =171 256 =16 (2)

Dividing the two equations we have =

32256 3 = 8 = 2

and 32 =13 32 =123 1 = 4

Thus,10 =11 = 4 2101 = 2048

The general term is =

and the sum =()





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5/7

Sigma Notation

Sigma notation is a way of expressing sums uses the Greek letter

=1

where is the index of summation taking values from 1to .

Properties of Sigma Notation

( ) =

=1

=1

=1

=

=1

=1

=1=

Useful formulas of Sigma Notation

=

=1

( + 1)2





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6/7

2

=1=

( + 1)(2 + 1)6

3

=1=

2( + 1)24

Binomial Theorem

( +) =

=0

where = !!()!

and !is the product of the first n positive integers for 1! = 1 2 3 and 0!=1

Example

Using the binomial theorem, expand (3 + 1)4

Solution

Using the formula

( +) =

=0

we have that





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7/7

(3 + 1)4 = = 4(3)414 =4=0 40(3)40 + 41(3)41 +

4

2(3

)42 +

4

3(3

)43 +

4

4(3

)44 = 4!

0!(40)!81

4 + 4!

1!(41)!27

3 +

+ 4!

2!(42)! 92 + 4!

3!(43)! 3 + 4!

4!(44)! = 814 + 1083 + 542 + 12 + 1





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sequences series and binomial theorem

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