Sequences,Strings.pptx

Sequences and Strings

Sequences

 A sequence is a special case of function, where domain of the function is a subset of

N={1,2,3,4 …}. Let s or {s_n} be a sequence of values s₁ , s₂ , s₃ , s₄ ,….we use the notation s_n to denote value s(n).

 Sequence of all positive even integers:

s₁ =2, s₂=4 s₃ =6 … s_n =2n …

 Depending on domain sequence can be finite or

Example

Consider the sequence b_n = (1)n.

{b_n} = 1, 1, 1, 1, …

{b_n} denotes an infinite sequence of 1’s and

Recognizing Sequences

_{Sometimes, you’re given the first few terms}

of a sequence, and you are asked to find the sequence’s generating function, or a

Recognizing Sequences

_{Examples: What’s the next number?}

_1,2,3,4,… _{1,3,5,7,9,…} _{2,3,5,7,11,...}

5 (the 5th smallest number >0)

11 (the 6th smallest odd number >0)

Sequence Notation

Given a sequence {a_n} defined by rule

a_n= n2 – 1, for all n1

values of the function are :

a₁=0, a₂= 3, a₃= 8, a₄=15 …

another notation of this sequence is {n2 – 1} n1

We call this representation explicit . Each value a_n can be computed directly for arbitrary value n1.

Sequence Notation

We can obtain member a_n of this sequence in a different form

a₁=0, a_n+1= a_n +2n + 1, for all n1

values of the sequence are the same as before :

a₁=0, a₂= 3, a₃= 8, a₄=15 …

We call this representation recurrent. Each value a_n can be computed from the previous value a_n-1 .

Sequence Notation

A recurrence relation for the sequence {a_n} is an

equation that expresses a_n in terms of one or more of the previous terms of the sequence, namely,

a₀, a₁, . . . , a_n−1, for all integers n with n ≥ n₀, where

n₀ is a nonnegative integer. A sequence is called a

solution of a recurrence relation if its terms satisfy the recurrence relation

Example

Let {a_n} be a sequence that satisfies the

recurrence relation a_n = a_n−1 + a_n−2 for n =2, 3, 4, . . . , and suppose that a₀ = 0 and a₁ = 1. What are a₂ , a₃, a₄ , a₅,?

_{We see from the recurrence relation that}

a₂ = a₁+a₀ = 1+0 = 1, a₃ = a₂ +a₁ = 1+1 = 2,

a₄ = a₃+a₂ = 2+1 = 3, a₅ = a₄ +a₃ = 3+2 = 5.

Sequence Notation

Given a sequence {a_m} defined on set MN a sequence {a_k} defined on set KM is a

subsequence of {a_m}.

_{Sequence of all positive even numbers is a}

subsequence of all positive numbers.

_{sequence 3,5,7,8 is a subsequence of}

2,3,3,4,5,6,7,8,9 , but it is not a subsequence of

Summation Notation

Given a sequence {a_n}, the summation of {a_n}

from j to k is written and defined as follows:

_{Here, i is called the index of summation.} Let {a_n}= 1,2,3,4

k j j k j i

i a a a

a   _  





10 4 3 2 1 : 4     

Generalized Summations

 For an infinite series, we may write:

 To sum a function over all members of a set

 X={x₁, x₂, …}:

 Or, if X={x|P(x)}, we may just write:

... ) ( ) ( : )

(  ₁  ₂ 





... :  _1 







i

i a a

Simple Summation Example

4

2 2 2 2

2

( 1) (2 1) (3 1) (4 1) (4 1) (9 1) (16 1) 5 10 17

32

i

i



               



More Summation Examples

_{An infinite series with a finite sum:}

_{Using a predicate to define a set of elements to}

sum over: 87 49 25 9 4 7 5 3

22 2 2 2

10 prime)

is (

2 _{        }



2 ₄1

2 1 1 0 0            



Finding the sum of 1,2,…n

_{Consider the sum (assume n is even):}

1+2+…+(n/2)+((n/2)+1)+…+(n-1)+n

_{n/2 pairs of elements, each pair summing to}

n+1, for a total of (n/2)(n+1)=n(n+1)/2.

…

n+1

n+1

Example: Geometric Series

_{A geometric series is a series of the form}

a, ar, ar2, ar3, …, ark, where a,rR.

_{The sum of such a series is given by:}

_{We can reduce this to closed form via clever}

manipulation of summations...

0 0

k k

i i

i i

S ar a r

 

Geometric Series closed formula

(1+r+r2+r3+…rk)(1-r)= (1-r+r-r2+r2+…-rk+1)=

=(1-rk+1)

(1+r+r2+r3+…rk)=(1-rk+1)/(1-r)=

=a(1-rk+1)/(1-r)

0 0

k k

i i

i i

S ar a r

 

Example: Infinite Geometric Series

Let x be a real number with |x| < 1. Find summation From the previous slide we take a=1 and r=x,

we have

Then we have Special case: = 2

Strings

_{A string over X, where X is a finite set , is a}

finite sequence of elements from X.

_{Example : Let X={a,b,c} then sequence}  ₌

b,a,a,c,c,c is a string. We use notation as

=baaccc or shortly =ba2c3.

_{The length of a string is the number of its}

Strings

_{A concatenation of strings}  _=ac2 and =ba2c3

is string = ac2ba2c3 ,||=||+||

_{A substring is a subsequence of a string where}

elements are consecutive.  =acc is a

substring of , but  =bcc is not a substring of