Sequences and Series
Let's see the definition of sequences and series first.A set of numbers arranged in a definite order according to some definite rule is called a sequence.or A sequence is a function whose domain is the set N of natural numbers.It is customary to denote a sequence by a letter 'a' and the image a(n) or t(n), n Î N under 'a' by an or tn.Indicated sum of the terms in a sequence is called a series. The result of performing the additions is the sum of the series.The indicated sum a1+a2+a3+.....+an of the terms of a sequence .Here a1,a2,a3,....an is called a series.
i) 1 + 4 + 7 + 10 + ... is a series in which first term is 1, second term is 4, third term is 7 and so on.
ii) 3 - 9 + 27 - 81 + ... is also a series in which the first term is 3, second term is -9, third term is 27 and so on.
A set of numbers arranged in a definite order according to some definite rule is called a sequence. or A sequence is a function whose domain is the set N of natural numbers. A sequence is called finite if the number of terms is finite. A finite sequence has always a last term.Examples:
2, 5, 8, 11, 14 …, 32
37, 33 …, 1A sequence is called infinite if the number of terms is infinite. An
infinite sequence has no last term. In this sequence, every term is
followed by a new term.
- (i) Let X be a set of numbers and f : Nn → X be a function, then the ordered set {f(1), f(2),...., f(n)} is called a finite sequence in X.
(ii) Let X be a set of numbers and f : N → X be a function, then the ordered set {f(1), f(2),....} is called
an infinite sequence in X.
- If {Tn} is a sequence, then the sum T1 +T2 + T3.... is called the series corresponding to the sequence {Tn}. A series is called finite or infinite according as the corresponding sequence is finite or infinite.
- There are three methods of describing a sequence:
(i) A sequence may be described by writing first few terms of a sequence till the rule for writing down the other
terms of the sequence become evident.
(ii) A sequence may be described by giving a formula for its nth term.
(iii) A sequence may be described by specifying its first few terms and a formula to determine the other terms
of the sequence in terms of its proceeding terms.
- A sequence is said to be a progression if its terms numerically increases (respectively decreases).
- (i) A sequence {Tn} is said to be an arithmetic progression (A.P) if there exists a number, say d such that Tn+1 - Tn = d, n ≥1
The constant number 'd' mentioned above is called the common difference of the corresponding A.P.
(ii) If 'a' and 'd' be the first term and common difference of the A.P.
(iii) If 'a' and 'd' be the first term and common difference of the A.P. {Tn}, then the sum of first n terms, Sn is
given by
The form (a) is used when common difference 'd' is known and the form (b) is used when the last term 'l' is
known.
(iv) T1 = S1 and for n > 1, we have Tn = Sn - Sn-1.
(v) If the sequence a, A1, A2,.....,An, b is an A.P., then the numbers A1, A2,.....,An are called the n arithmetic
means between a and b.
(vii) The sum of n A.M.s between given numbers a and b is equal to n times the A.M. between a and b.
(viii) If a, b, c are in A.P., then for any k:
(a) a+k, b+k, c+k are in A.P.
(b) a-k, b-k, c-k are in A.P.
(c) ka, kb, kc are in A.P.
(d) a/k, b/k, c/k are in A.P. (k¹0).
(ix) (a) If the sum of three numbers in A.P. is given, then the numbers should be taken as a-d, a, a+d.
(b) If the sum of four numbers in A.P. is given, then the numbers should be taken as a-3d, a-d, a+d,
a+3d.
- (i) A sequence {Tn} of non-zero terms is said to be a geometric progression (G.P.) if there exists a number, say, r such that
The constant number 'r' mentioned above is called the common ratio of the corresponding G.P.
(ii) If 'a' and 'r' be the first term and common ratio of the G.P.
(iii) If 'a' and 'r' be the first term and common ratio of the G.P. {Tn}, then the sum of first n terms, Sn is given
by
These formulae are used when 'last term' is given.
(v) If 'a' and 'r' be the first term and common ratio of a G.P. such 
(vi) If the sequence a, G1, G2,....,Gn, b of positive numbers is a G.P., then the numbers G1, G2,....,Gn, are
called the n geometric means between a and b.
(vii) The G.M. between given positive numbers a and b is equal to 
(viii) The product of n G.M.s between given positive numbers a and b is equal to nth power of the G.M.
between a and b.
(ix) If a, b, c are in G.P, then for any non-zero k,
(a) ka, kb, kc are in G.P.
(b) a/k, b/k, c/k are in G.P.
(x) (a) If the product of three numbers in G.P. is given, then the numbers should be taken as a/r, a, ar.
(b) If the product of four numbers in G.P. is given, then the numbers should be taken as a/r3, a/r, ar, ar3
- (i) A sequence of non-zero numbers is said to be a harmonic progression (H.P.) if the sequence of the reciprocals of its terms is an A.P.
(iii) There is no formula to find the sum of first n terms of a H.P.
(iv) If the sequence a, H1, H2,.....,Hn, b of non-zero numbers is a H.P., then the numbers H1, H2,.....,Hn are
called n H.M.s between a and b.
- If A, G, H are the A.M., G.M., H.M. respectively between non-zero positive numbers a and b, then
(a) A, G, H are in G.P.
(b) A>G>H. In particular, if a = b = c, then A = G = H.
- (i) A sequence is said to be an arithmeticco-geometric sequence if terms of the sequence are the products of corresponding terms of an A.P. and a G.P.
(ii) For the A.G. sequence a, (a+d)r, (a+2d)r2,…... We have
(iii) If for the A.G. sequence a, (a+d)r, (a+2d)r2,....., then the sum up to infinity, S, given by
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| More topics in Sequences and Series | |
| Progression | Sequence |
| Series | Exponential Series |
| Logarithmic Series | Taylor Series Expansion |
| Sequence and Number Patterns | |
