Basic Concepts of Sequence and Series: 

The word sequence is used in a similar way in mathematics as that in English i.e. in an ordered form. Let us consider a situation where we have 5 boys with an age difference of two years, where age of younger boy is 12yrs, there we will form a sequence. The various numbers occurring in a sequence is called its terms. The terms in the sequence are denoted by a₁ a₂

a₃ a₄ ........ a_n etc where subscripts denote the position of terms. The nth term is the nth position of sequence, denoted by a_n. The nth term is also known as general term.

Thus the terms of the sequence of boy's age mentioned above are:

a₁ = 12, a₂= 14, a₃ = 16, a₄ = 18 and a₅ = 20

Where a₁ is the first term of sequence.

Now, sequence can be categorized into two types as finite sequence and infinite sequence. Finite Sequence is the one's which have finite number of terms and Infinite sequence is the one's which are not finite.

A sequence is a function whose domain is set of natural numbers denoted by a(n) or a_n.

If a₁ a₂  a₃  a₄ ........ a_n be a given sequence, then the expression a₁ + a₂  + a₃ +  a₄ +... a_n+.... Is the series associated with the given sequence? The series is finite or infinite depends on the sequence. The series is also denoted by sigma notation (∑).

Arithmetic Progression (AP) 

Arithmetic progression (AP) or math succession is a sequence in which terms increment or lessening consistently by same constant D (d is known as normal distinction)

An arithmetic progression is given by an, (A+ D), (A + 2D), (A + 3D), ... ..

Where A is the principal term and D is the common difference of the AP.

The general term i.e. nth term of AP is given by a = A+ (n-1)D

Number of terms of an AP is given by n = (L-A)D + L, where n = number of terms, A= the primary term , L = last term, D= common difference

Entirety of first n terms in an AP is given by Sn = n/2 [ 2A+(n-L)D ] = n/2 (A+ L)

where A = the first term, D= common difference, L = tn = nth term = A + (n-1)D

Arithmetic Mean A of any two numbers a and b is given by C+D/2 i.e. the sequence of C, A and D is in A.P.

Note: If each term of AP is included, subtracted, increased or partitioned by the same non-zero steady, the subsequent grouping likewise will be in AP.

Geometric Progression:

If the ratio of any term to it preceding term is same throughout, then that sequence is known as Geometric Progression or G.P. Here the constant factor is known as common ratio.

A geometric progression(GP) is given by A, AR, AR², AR³, ... where A = the first term , AR = second term, R = the common ratio

The general term i.e. nth term of G.P. is given by A_n = AR^n-1

  Sum of first n terms in an GP is given by S_n = A(Rⁿ- 1) / (R-1) if R > 1 or S_n = a(1-Rⁿ) / (1-R) if         R<1, where A = the primary term, R = basic proportion , n = number of terms.

Sum of an infinite G.P. is given by S∞=A1-R  (if -1 < R < 1), where A = the primary term, R = basic proportion

Geometric Mean (G.M.) of any two positive numbers a and b is given by √CD i.e. the sequence C, G, D is G.P.

Sequence a₁ = 4  , a₂= 16, a₃ = 64, a₄ = 256 where first term is 4, second term ar is 16 common ratio (r) = 4.

Note: If C, G, D are in GP, C-G / G-D =CG

Harmonic Progression:

Non-zero numbers A₁, A₂, A₃,.........A_n  are in Harmonic Progression (HP) if 1 / A₁, 1 / A₂, 1 / A₃..... 1 / A_n are in AP. HP is also known as harmonic sequence.

The general term i.e. n^th term of the HP = 1 / A+(n-1)D

Harmonic Mean (HM) B when A, B and C are in Harmonic sequence is given by B = 2AC / (A+C)

Note: If A, B, C are in HP, 2 / B = 1 / A + 1 / C

Relationship between AM, HM, and GM of Two Numbers:

Let GM, AM and HM be the Geometric Mean, Arithmetic Mean and Harmonic Mean of two positive numbers respectively, then GM² = AM × HM

Sum to n terms of Special Series:

Sum of the first n natural numbers = 1 + 2 + 3 + ......+n =∑n = n(n+1) / 2

Sum of squares of the first n natural numbers = 1² + 2² + 3² + .....+ n² = ∑n² = n(n+1)(2n+1) / 6

Sum of cubes of the first n natural numbers =  1³ + 2³ + 3³ +...+ n³=∑n³ = n²(n+1)² / 4 =                  [ n (n+1) / 2]²

Importance of Sequence and Series:

On the off chance that you glance around in your environment, you will locate various examples in nature - leaves and blooms with comparable structures, the swells on a lake, the symmetry of a starfish and numerous more examples that don't stop to astonish us. Nature enlivened mathematicians to attempt and clarify these examples in nature; to deal with numerical models and comprehend the nuts and bolts of geometric shapes and structures. A great deal of work has been done in the field of number sequences and series to anticipate the likelihood of an occasion, planning structures and structures, investigation of genuine circumstances and so forth. Throughout the centuries, legends have created around numerical issues including sequences and series. A standout amongst the most well-known legends about sequences and series concerns the innovation of chess. Archaeologists in the continuous or motion pictures can anticipate the period of various curios by utilizing sequences and series.

Sequences and series are likewise utilized as a part of the business and money-related examination to aid basic leadership and locate the best answer for a given issue. (To comprehend the scientific portrayals and subtle elements of number sequence and series formulas, take this essential course on measurements and likelihood.) Organizations utilize quantitative investigation in hazard appraisal and administration, settling on speculation choices, evaluating and numerous more vital capacities. On the off chance that you are a business expert, an analyst or a speculation director, your work will spin around number examples and the investigation of these examples.

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