Sequence

A sequence may be defined as :
An ordered collection of numbers a₁,a₂,a₃--- a_n---- is a sequence if according to some definite rule or law, there is a definite value of a_n called the term or element of the sequence, corresponding to any value of the natural number n.

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Clearly a₁= 1^stterm, a₂ = 2^ndterm, a_n= n^th term, of a sequence.
Thus it is clear that the n^th term of a sequence is a function of the positive integer n. The n^th term is also known as the general term of the sequence.

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A sequence may be finite or infinite.

A finite sequence a₁,a₂,a₃---a_n is denoted by {a_i}ⁿ_i=1 and an infinite sequence a₁,a₂,a₃---a_n is denoted by {a_n}_n=1 or simply by {a_n}, where a_n is the n^th element of the sequence.
e.g.
(i)25,3,6,31,------
(ii) 2,11,17,43,61,------
(iii) 2,4,6,8,------
(iv) 10,20,30,40,------
(i) and (ii) are not arranged in particular order (Do not follow any rule or law). So the numbers in the collection (i) and (ii) do not form sequence. While , (iii) and (iv) are arranged in a particular order (which obey some definite rule or law). So the numbers in the collections (iii) and (iv) form sequences.

e.g. Infinite sequences.
(i) {1/n} is 1, 1/2, 1/3,1/4----
(ii) {(-1)ⁿ.n} is -1,2,-3,4,-----
(iii) {(n+1)/(n+2)} is 2/3, 3/4,4/5,------

e.g. Finite sequences.
(i) A sequence of even positive integers with in 20 is
2,4,6,8,10,12,14,16,18.
(ii) A sequence of odd positive integers upto 10 is.
1,3,5,7,9

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Arithmetic progression

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Important concepts :

(a) . Two terms in AP = a-d,  a+d

(b) . Three terms in AP = a-d,  a,  a+d

(c) . Four terms in AP =  a-3d,  a-d,  a+d,  a+3d

(d) . Five terms in AP = a-2d,   a-d,   a,  a+d,  a+2d

(e) . T_n = S_n - S_n-1

(f) . d = T_n - T_n-1

(g) . S₁ = a(first term)

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Geometric Progression

In Geometric Progression there is a common ratio between two consecutive terms,common ratio is denoted by r.

The structure of geometric series is ,

a,   ar,   ar²,    ar³---------------





Sum of Finite Geometric Progression



Important Concepts:

(a) Two Terms in GP  = a/r,   ar

(b) Three Terms in GP = a/r,   a,   ar

(c) Four Terms in GP =  a/r³,   a/r,   ar,   ar³

(d) Five Terms in GP =  a/r²,   a/r,   a,   ar,   ar²

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Sum of Infinite Terms