A Sequence is a method of natural numbers if its code range consists of a series of complex or real numbers. A sequence can be finite depending on whether its representatives are finite or infinite. We could also define progression as a set of conditions in a particular model. Furthermore, a series is formed by adding or removing series members. A Series is called a limited series if it has a bounded number of members, whilst an infinite series has an unlimited number of members. A pattern is a list of objects that have been organized in a specific order. As the total of all terms in a sequence, a series can be strongly generalized. However, there must be a clear relationship between all of the sequence’s terms.
|Table of Contents|
|Sequences of Arithmetic|
|Sequences of Geometry|
|Sequences of Harmonics|
|Properties of Fibonacci Sequences|
|Application of Fibonacci Sequences|
|Finite and Infinite|
Sequence and Series Types
Among the most common sequence examples are:
- Sequences of Arithmetic
- Sequences of Geometry
- Sequences of Harmonics
- Fibonacci Sequences
Sequences of Arithmetic
An arithmetic sequence is one in which each term is formed by adding and subtracting a specific number from the last number. The arithmetic Sequence formula is used to find every term in the arithmetic sequence. For calculating the nth term of an algebraic expression, use the arithmetic sequence formula. The arithmetic sequence is the series in which the common difference of the two successive terms remains unchanged.
Arithmetic Formula: an = a1 + (n – 1)d
- an= Nth term
- a1 = 1st term of the sequence
- n = Number of terms
- d = Common difference
- Find the value of the 25th term of the Arithmetic Sequence of 2,6,10,14….
- The given sequence is 2, 6, 10, 14…
First Term, a= 2
Common Difference, d= 6-2= 4
Using the sequence and series formula
an= a+ (n-1) d
For 25th term, Substitute n = 25
a25= a+ 24d = 2+ 24X 4= 2+ 96= 98
- Find the sum of the first 100 terms of series 1+ 5+9+….
- a= 1 and d= 4
Formula= Sn= n/2 (2a+ (n-1) d)
100/ 2 (2 (1)+ (100-1) 4)
Sequences of Geometry
A geometric sequence is one in which each term is acquired by multiplying or separating a definite number by the preceding number. A geometric Sequence is a number series wherein the ratio of each and every 2 sequential numbers has always been the same.
Geometric Sequence Formula: Nth term: an = a · rn – 1
Sum of n term Geometric Formula: Sn = a + ar + ar2 + … + arn-1
Sum of N terms Sn = a (rn – 1) / (r – 1), when r > 1 (or) when r < -1
Infinite Geometric Sequence Formula: a, ar, ar2, ar3, …. is, S∞ = a / (1 – r)
- Find the 10th term of the sequence 1, 2, 4, 8….
- The first term is, a= 1
Common ratio is 2/1 = 4/2 = 8/4
Using formula for nth term of sequence is,
10th term, n=10
Consider the sequence 1, 3, 5,7,9,11. Find the common ratio and 9th term.
Common Ratio, r = 3/1 = 3
Term is multiplied by 3 to obtain next term
Nth term of sequence is Tn and given is Tn= ar (n-1)
Here a= 1, r= 3 and 9= 9
Formula= T9= 1X (3)9-1
Sequences of Harmonics
A number series is indeed a chromatic series if the reciprocals of all its elements establish an arithmetic sequence. Harmonic Progression is obtained by combining the reciprocal of an arithmetic progression’s terms. If the provided arithmetic progression concepts are a, a + d, a + 2d, a + 3d,…Therefore the harmonic progression concepts are 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d), 1/(a + 4d),……
Formula: nth term of Harmonic Sequence
= 1/(a + (n – 1)d)
That is equal to the inverse of an arithmetic progression’s nth term. The reciprocal of a total of first term, as well as the (n – 1) times of a common difference, is the nth term of a harmonic progression. The nth term can be used to find any one of the harmonic sequence terms.
Harmonic Mean Formula: HM = n / [1/x1 + 1/x2 + 1/x3 + … + 1/xn]
Application of Sequence
Harmonic means have many applications in mathematics, technology, quantum mechanics, and enterprise. The Harmonic Mean of a corresponding speed can be used to calculate the mean speed over two sets of equal spacing. If the vehicle’s speed is x mph for the first d miles and y mph for the next d miles, the vehicle’s average speed over the full distance is equivalent to the chromatic mean of these two speeds. (2xy) / (x + y) = Average Speed. The density values of the individual components can be used to measure the density of a combination or the density of a composite material of two substances of uniform density and percentage composition. The profit earning factor is computed within the area of accounting by using the theory of a weighted harmonic average of individual components.
- Determine the value of 21st term of harmonic progression of ½, ⅙, 1/10, 1/11….
- Given sequence= ½, ⅙, 1/10, 1/11…
= 1/a, 1/ (a+d), 1/ (a+2d)…
= 1/a+ 1/ 2, 1/ (a+d)= ⅙
= a2, a+d= 6 or d= 6
21st term= 1/ (a+20d)= 1/ ( 2+ 20×4)
= 1/ (2=80)
Nth term= 1/ (a+ (n-1) d)
= 1/ (2+ (n-1)4)
=1/ (2+ 4n-4)
= 1/ (4n+ 1)
Therefore, 21st term is 1/ 82 and nth term is 1 (4n+ 1)
Fibonacci Sequence is a series of numbers where an amount is the sum of the two numbers, commencing with 0 and 1. Fibonacci sequence states that each number in the sequence is the sum of the two numbers previous to it in the series.
Formula: Fn = Fn-1 + Fn-2, where n > 1
Properties of Fibonacci Sequences
The golden ratio is related to Fibonacci numbers. The golden ratio can be used to calculate any Fibonacci number, Fn =(n – (1-)n)/5, and where’s the golden ratio as well as 1.618034. Each nth number would be numerous of the number n. Examine the sequence for another intriguing pattern. The third number in the sequence is a multiple of two. Every fourth number in the sequence is numerous of three, while every fifth number is a multiple of five.
Application of Fibonacci Sequences
The Fibonacci sequence could be found in various fields, including nature, songs, and physical figures. Fibonacci Sequences are used in number clusters and a fantastic percentage in music these days. Coding, computerized systems, interlinking parallel, and distributed systems are used for many areas of science, such as highly energetic hard science, quantum theory, cryptography, and so forth.
- Find the 12th term of the Fibonacci Sequence. If the 10th and 11th terms are 24 and 34.
- 12th term is the sum of the 10th and 11th term
12th term= 10th + 11th term
12th term= 58
- Find F14 in the Fibonacci Sequence i.e. 244. Find the next term.
- F15= F14x Golden Ratio
F15= 244X 1.618034
Arithmetical scenes and sequences can also be used in accounting and business analysis to aid in judgment and determine the optimal method for a specific issue. Institutions use quantification for risk evaluation, investment decisions, pricing, and various other features. A sequence is considered a collection of numbers and objects that follow a specific pattern. Whenever the sequence’s items are added around each other, they consist of a series. Arithmetic Progression is a series in which the consecutive terms have a common difference, such as 2, 4, 6, 8, etc.
1. What is the difference between Series and Sequence.
Series and Sequences are different aspects. Series is the addition of the terms of the sequence (an). A series, like a sequence, can be finite or infinite, with such a finite number of words issued as a1 + a2 + a3 + a4 + a5 + a6 +……an. In comparison to infinite numbers, where the number of components is not finite or is infinite, recorded as a1 + a2 + a3 + a4 + a5 + a6 +……an +….
Only when a1 + a2 + a3 + a4 + a5 + a6 +…… a = Sn, then Sn is the sum of the sequence’ n elements. The quantity of aspects is frequently denoted by the Greek alphabet sigma. The representatives of the sequential manner are referred to as terms or elements, and then they can be any natural numeric value. Each term of such a sequence is related to the term which precedes and follows it. Overall, sequences have concealed rules or patterns which help you figure out all the values of the next term. This same nth term is indeed the feature of numeric n (positive), and it is considered the sequence’s general term. A sequence can be either finite or infinite.
2. What is the Finite and Infinite sequence?
If a sequence has a finite term, it is finite; otherwise, it is infinite. 4,8,12,16,…,64 is a finite sequence. The first term in the series is four, and the final term is 64. The sequence is finite because it has the last term. 4,8,12,16,20,24,… is an infinite sequence. The sequence’s first term is 4. It indicates that the sequence continues indefinitely; there is no end term. It is a never-ending sequence.
3. If 4, 6, 8, 10, 12, 14, 16….Find the common differences, the nth term, and 21st term
Sequence is 4, 6, 8, 10, 12, 14, 16
Common difference= 6-4= 2
The nth term of arithmetic sequence is denoted as the term Tn and is given as Tn= a+ (n-1)d
Here “ a” is the first term and d is the difference
Tn= 4+ (n-1) 2
= 4+ 2n-2= 2n+1
Now, 21st term= T21= 4+ (21-1) 2
4. List important formulas of Sequence and Series.
- A. nth term of the arithmetic sequence or arithmetic progression (A.P) is given by an = a + (n – d)
- The arithmetic mean [A.M] between a and b is A.M = [a + b] / 2
- The nth term an of the geometric succession or geometric progression [G.P] is an = a * rn–1
- The nth term an of the harmonic progression is an= 1 / [a + (n – 1) d]
- The harmonic mean between a and b is H.M = 2ab / [a + b] Series of A.P : Sn = [n / 2] (a + l), Sn = (n / 2) [2a + (n – 1) d], where Sn is sum to n terms of A.P.
- The sum of ‘n’ A.M between a and b is A.M = [n (a + b)] / 2
- Series of G.P: Sn = [a (1 – rn)] / [1 – r] ; where Sn is the sum to n terms of G.P.
- The sum of infinite geometric series is S = a / 1–r.
5. What are the properties of Arithmetic Progression?
An Arithmetic Progression is a set of numbers where every other term is acquired by attaching a set number to a number before it. The first term is a, the second term is a common difference, the third term is an, and the complete number of terms is n. AP can be written as a, a+d, a+2d, and so on, as well as the nth term of an AP, can be estimated as a = a + (n1)d.
An AP’s sum could be calculated as sn=n/2[2a+(n1)d].
The slope of a straight line serves as the following differences in an AP graph.
The shared difference does not have to be a positive one. For instance, consider the widely varied in the series 16,8,0,8,16,…