Examples and notation Sequence

1 examples , notation

1.1 examples
1.2 indexing
1.3 defining sequence recursion

examples , notation

a sequence can thought of list of elements particular order. sequences useful in number of mathematical disciplines studying functions, spaces, , other mathematical structures using convergence properties of sequences. in particular, sequences basis series, important in differential equations , analysis. sequences of interest in own right , can studied patterns or puzzles, such in study of prime numbers.

there number of ways denote sequence, of more useful specific types of sequences. 1 way specify sequence list elements. example, first 4 odd numbers form sequence (1, 3, 5, 7). notation can used infinite sequences well. instance, infinite sequence of positive odd integers can written (1, 3, 5, 7, ...). listing useful infinite sequences pattern can discerned first few elements. other ways denote sequence discussed after examples.

examples

a tiling squares sides successive fibonacci numbers in length.

the prime numbers natural numbers bigger 1 have no divisors 1 , themselves. taking these in natural order gives sequence (2, 3, 5, 7, 11, 13, 17, ...). prime numbers used in mathematics , in number theory.

the fibonacci numbers integer sequence elements sum of previous 2 elements. first 2 elements either 0 , 1 or 1 , 1 sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...).

for large list of examples of integer sequences, see on-line encyclopedia of integer sequences.

other examples of sequences include ones made of rational numbers, real numbers, , complex numbers. sequence (.9, .99, .999, .9999, ...) approaches number 1. in fact, every real number can written limit of sequence of rational numbers, e.g. via decimal expansion. instance, π limit of sequence (3, 3.1, 3.14, 3.141, 3.1415, ...). related sequence sequence of decimal digits of π, i.e. (3, 1, 4, 1, 5, 9, ...). sequence not have pattern discernible eye, unlike preceding sequence, increasing.

indexing

other notations can useful sequences pattern cannot guessed, or sequences not have pattern such digits of π. 1 such notation write down general formula computing nth term function of n, enclose in parentheses, , include subscript indicating range of values n can take. example, in notation sequence of numbers written



(
2
n

)

n
∈

n





{\displaystyle (2n)_{n\in \mathbb {n} }}

. sequence of squares written



(

n

2



)

n
∈

n





{\displaystyle (n^{2})_{n\in \mathbb {n} }}

. variable n called index, , set of values can take called index set.

it useful combine notation technique of treating elements of sequence variables. yields expressions



(

a

n



)

n
∈

n





{\displaystyle (a_{n})_{n\in \mathbb {n} }}

, denotes sequence nth element given variable




a

n




{\displaystyle a_{n}}

. example:

a

1





=
1

st element of 

(

a

n



)

n
∈

n








a

2





=
2

nd element 






a

3





=
3

rd element 







⋮





a

n
−
1





=
(
n
−
1
)

th element






a

n





=
n

th element






a

n
+
1





=
(
n
+
1
)

th element







⋮






{\displaystyle {\begin{aligned}a_{1}&=1{\text{st element of }}(a_{n})_{n\in \mathbb {n} }\\a_{2}&=2{\text{nd element }}\\a_{3}&=3{\text{rd element }}\\&\vdots \\a_{n-1}&=(n-1){\text{th element}}\\a_{n}&=n{\text{th element}}\\a_{n+1}&=(n+1){\text{th element}}\\&\vdots \end{aligned}}}

note can consider multiple sequences @ same time using different variables; e.g.



(

b

n



)

n
∈

n





{\displaystyle (b_{n})_{n\in \mathbb {n} }}

different sequence



(

a

n



)

n
∈

n





{\displaystyle (a_{n})_{n\in \mathbb {n} }}

. can consider sequence of sequences:



(
(

a

m
,
n



)

n
∈

n




)

m
∈

n





{\displaystyle ((a_{m,n})_{n\in \mathbb {n} })_{m\in \mathbb {n} }}

denotes sequence mth term sequence



(

a

m
,
n



)

n
∈

n





{\displaystyle (a_{m,n})_{n\in \mathbb {n} }}

.

an alternative writing domain of sequence in subscript indicate range of values index can take listing highest , lowest legal values. example, notation



(

k

2



)

k
=
1


10




{\displaystyle (k^{2})_{k=1}^{10}}

denotes ten-term sequence of squares



(
1
,
4
,
9
,
.
.
.
,
100
)


{\displaystyle (1,4,9,...,100)}

. limits



∞


{\displaystyle \infty }

,



−
∞


{\displaystyle -\infty }

allowed, not represent valid values index, supremum or infimum of such values, respectively. example, sequence



(

a

n



)

n
=
1


∞




{\displaystyle (a_{n})_{n=1}^{\infty }}

same sequence



(

a

n



)

n
∈

n





{\displaystyle (a_{n})_{n\in \mathbb {n} }}

, , not contain additional term @ infinity . sequence



(

a

n



)

n
=
−
∞


∞




{\displaystyle (a_{n})_{n=-\infty }^{\infty }}

bi-infinite sequence, , can written



(
.
.
.
,

a

−
1


,

a

0


,

a

1


,

a

2


,
.
.
.
)


{\displaystyle (...,a_{-1},a_{0},a_{1},a_{2},...)}

.

in cases set of indexing numbers understood, subscripts , superscripts left off. is, 1 writes



(

a

k


)


{\displaystyle (a_{k})}

arbitrary sequence. often, index k understood run 1 ∞. however, sequences indexed starting zero, in

(

a

k



)

k
=
0


∞


=
(

a

0


,

a

1


,

a

2


,
.
.
.
)
.


{\displaystyle (a_{k})_{k=0}^{\infty }=(a_{0},a_{1},a_{2},...).}

in cases elements of sequence related naturally sequence of integers pattern can inferred. in these cases index set may implied listing of first few abstract elements. instance, sequence of squares of odd numbers denoted in of following ways.

(
1
,
9
,
25
,
.
.
.
)


{\displaystyle (1,9,25,...)}






(

a

1


,

a

3


,

a

5


,
.
.
.
)
,


a

k


=

k

2




{\displaystyle (a_{1},a_{3},a_{5},...),\qquad a_{k}=k^{2}}






(

a

2
k
−
1



)

k
=
1


∞


,


a

k


=

k

2




{\displaystyle (a_{2k-1})_{k=1}^{\infty },\qquad a_{k}=k^{2}}






(

a

k



)

k
=
1


∞


,


a

k


=
(
2
k
−
1

)

2




{\displaystyle (a_{k})_{k=1}^{\infty },\qquad a_{k}=(2k-1)^{2}}






(
(
2
k
−
1

)

2



)

k
=
1


∞




{\displaystyle ((2k-1)^{2})_{k=1}^{\infty }}

moreover, subscripts , superscripts have been left off in third, fourth, , fifth notations, if indexing set understood natural numbers. note in second , third bullets, there well-defined sequence



(

a

k



)

k
=
1


∞




{\displaystyle (a_{k})_{k=1}^{\infty }}

, not same sequence denoted expression.

defining sequence recursion

sequences elements related previous elements in straightforward way defined using recursion. in contrast definition of sequence elements function of position.

to define sequence recursion, 1 needs rule construct each element in terms of ones before it. in addition, enough initial elements must provided subsequent elements of sequence can computed rule. principle of mathematical induction can used prove in case, there 1 sequence satisfies both recursion rule , initial conditions. induction can used prove properties sequence, sequences natural description recursive.

the fibonacci sequence can defined using recursive rule along 2 initial elements. rule each element sum of previous 2 elements, , first 2 elements 0 , 1.

a

n


=

a

n
−
1


+

a

n
−
2




{\displaystyle a_{n}=a_{n-1}+a_{n-2}}

,    




a

0


=
0


{\displaystyle a_{0}=0}

,




a

1


=
1


{\displaystyle a_{1}=1}

.

the first ten terms of sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, , 34. more complicated example of sequence defined recursively recaman s sequence. can define recaman s sequence by

a

0


=
0


{\displaystyle a_{0}=0}

,




a

n


=

a

n
−
1


−
n


{\displaystyle a_{n}=a_{n-1}-n}

, if result positive , not in list. otherwise,




a

n


=

a

n
−
1


+
n


{\displaystyle a_{n}=a_{n-1}+n}

.

not sequences can specified rule in form of equation, recursive or not, , can quite complicated. example, sequence of prime numbers set of prime numbers in natural order, i.e. (2, 3, 5, 7, 11, 13, 17, ...).

many sequences have property each element of sequence can computed previous element. in case, there function f such n,




a

n
+
1


=
f
(

a

n


)


{\displaystyle a_{n+1}=f(a_{n})}

.