Sequence & Series

Sequence & Series

Introduction:

A sequence is a set of terms in a definite order with a rule of obtaining its terms. Let N = {1, 2, 3, ...} be a set of positive consecutive integers.

In general, a sequence may be defined as a mapping f : N X, where X is a non empty set. The elements of the range set X are denoted by a₁, a₂, ...,a_n, ..., in which a_n is the

general term of the sequence. We write the sequence as {a_n} or {a₁, a₂, a₃, ..., a_n, ...}.

There is a one-to-one correspondence between the elements of the domain set N and the range set X. If the domain set N has a finite number of elements, then the range set X has also

a finite number of elements and the sequence {a₁, a₂, ..., a_n} is called a finite sequence. If the domain set N has infinite number of elements, then the range set also has an infinite

number of elements and we call the sequence an infinite sequence.

The sum a₁ + a₂ + a₃ + ... + a_n + ..., of the elements of the sequence {a_n} is called a series. a₁ is called the first term of the series, a₂ is called the second term of the series and a_n is the

nth term or the general term of the series. The series is finite if the sequence {a_n} is finite and the series is infinite if the sequence is infinite. If the term of a sequence follow a certain

pattern, then the sequence is called a progression.



Arithmetic Progression (A.P.):

A sequence of numbers is said to be in A.P. when the difference between any two consecutive numbers is always same. If a is the first term and d the common difference, the A.P. can

be written as a, a + d, a + 2d,...., a + (n - 1)d. The nth term t_n is given by t_n = a + (n - 1)d. The sum S_n of the first n terms of such an A.P. is given by



where l is the last term (i.e. the nth term of the A.P.).


Notes:
1. If a fixed number is added (subtracted) to each term of a given A.P. then the resulting sequence is also an A.P. with the same common difference as that of the given A.P.

2. If each term of an A.P. is multiplied by a fixed number(say k) (or divided by a non-zero fixed number), the resulting sequence is also an A.P. with the common difference multiplied by k.

3. If a₁, a₂, a₃,....and b₁, b₂, b₃,....are two A.P.'s with common differences d and d' respectively then a₁ + b₁, a₂ + b₂, a₃ + b₃......, is also an A.P. with common difference d + d'. If a_n

and b_n are the nth terms of the two A.P.'s, then the nth term of the resulting A.P. is a_n + b_n = a + (n - 1)d + b + (n - 1)d' = (a + b) + (n - 1) (d + d').


4. If we have to take three terms in an A.P., it is convenient to take them as a - d, a,a + d. In general, we take a - rd, a - (r - 1)d, ...., a - d, a, a + d, ...., a + rd, in case we have to

take (2r + 1) terms in an A.P.

5. If we have to take four terms, we take a - 3d, a - d, a + d, a + 3d. In general, we take a-(2r - 1)d, a - (2r - 3)d,...., a - d, a + d, ...., a + (2r - 1)d, in case we have to take 2r terms

in an A.P.

6. If a₁, a₂, a₃, ..., a_n are in A.P., then a₁ + a_n = a₂ + a_n-1 = a₃ + a_{n -2} = .... and so on.



Arithmetic Mean(s):

1. If three terms are in A.P., then the middle term is called the arithmetic mean (A.M.) between the other two i.e. if a,b,c are in A.P. then is the A.M. of a and c.

2. If a₁, a₂,...., a_n are n numbers then the arithmetic mean (A) of these numbers is

3. The n numbers A₁, A₂, .... , A_n are said to be A.M.'s between the numbers a

and b if a, A₁, A₂, ...., A_n, b are in A.P. If d is the common difference of this A.P. then b = a + (n + 2 - 1)d

.



Geometric Progression (G.P.)

A G.P. is a sequence whose first term is non-zero and each of whose succeeding term is r times the preceding term, where r is some fixed non-zero number, known as the common

ratio of the G.P. If a is the first term of a G.P. and r its common ratio, then its nth term t_n is given by t_n = ar^n-1. The sum S_n of the first n terms of the G.P. is





Notes:

1. If each term of a G.P. is multiplied (divided) by a fixed non-zero constant, then the resulting sequence is also a G.P. with same ratio as that of the given G.P. .

2. If each term of a G.P. (with common ratio r) is raised to the power k, then the resulting sequence is also a G.P. with common ratio r^k.

If the given G.P. is a, ar, ar², ......... and each term is raised to the kth power, we obtain the sequence a^k, a^kr^k, a^kr^2k, ....... which is also a G.P. with a^k as the first term and r^k as the

common ratio.

1. If a₁, a₂, a₃,.... and b₁, b₂, b₃,.... are two G.P.'s with common ratios r and r' respectively then the sequence a₁b₁, a₂b₂, a₃b₃, .... is also a G.P. with common ratio rr'.

2. If we have to take three terms in a G.P., it is convenient to take them as a/r, a, ar.

In general, we take in case we have to take (2k + 1) terms in a G.P.

3. If we have to take four terms in a G.P., it is convenient to take them as a/r³, a/r, ar, ar³. In general, we take, in case we have to take 2k

terms in a G.P.

4. If a₁, a₂,.... ,a_n are in G.P., then a₁a_n = a₂ a_n-1 = a₃ a_n-2 = ....

5. If a₁, a₂, a₃,.... is a G.P. (each a_I > 0), then loga₁, loga₂, loga₃ .... is an A.P. The converse is also true.



Geometric Mean(s):

1. If three terms are in G.P., then the middle term is called the geometric mean (G.M.) between the two. So if a, b, c are in G.P. then b =is the geometric mean of a and c.

2. If a₁, a₂,...., a_n are positive numbers then their G.M.(G) is given by G = (a₁a₂a₃....a_n)^1/n.

3. If G₁, G₂,....G_n are n geometric means between a and b then a, G₁, G₂,...., G_n, b will be a G.P. Here b = ar^{n + 1}

r = G₁ = a, G₂ = a,.... , G_n = a.




Arithmetico-Geometric Progression

Suppose a₁, a₂, a₃,.... is an A.P. and b₁, b₂, b₃,.... is a G.P. Then the sequence a₁b₁, a₂b₂,...., a_nb_n is said to be an arithmetico-geometric progression. An arithmetico-geometric

progression is of the form ab, (a + d)br, (a + 2d)br², (a + 3d)br³,....

Its sum S_n to n terms is given by

S_n = ab + (a + d)br + (a + 2d)br² +....+(a + (n - 2)d)br^n-2 + (a + (n - 1)d)br^n-1.


Multiply both sides by r, so that

rS_n = abr + (a + d)br²+.... +(a + (n - 3)d)br^{n- 2} + (a + (n - 2)d)br^{n- 1} + (a + (n - 1)d)brⁿ.

Subtracting we get

(1 - r)S_n = ab + dbr + dbr² +.......+ dbr^{n - 2} + dbr^{n - 1} - (a + (n - 1)d)brⁿ

=

.


If -1 < r < 1, the sum of the infinite number of terms of the progression is .

1. (1 - x)^-1 = 1 + x + x² + x³ +.... - 1 < x < 1.

2. (1 - x)^-2 = 1 + 2x + 3x² +.... - 1 < x < 1.



Harmonic Progression (H.P.)

The sequence a₁, a₂, a₃,...., a_n,.... (a_i 0) is said to be in H.P. if the sequence is in A.P. The n^th term of the H.P. is , where





Harmonic Mean(s):

1. If a and b are two non-zero numbers, then the harmonic mean of a and b is a number H such that the numbers a, H, b are in H.P. We have

2. If a₁, a₂,....,a_n are n non-zero numbers, then the harmonic mean H of these number is given by H = .

3. The n numbers H₁, H₂,....,H_n are said to be harmonic means between a and b, if


a, H₁, H₂,.... ,H_n, b are in H.P. i.e if are in A.P. Let d be the common difference of the corresponding A.P., Then

d =.

Thus .



Method of Differences

Suppose a₁, a₂, a₃,.... is a sequence such that the sequence a₂ - a₁, a₃ - a₂,.... is either an A.P. or a G.P.. The n^th term, a_n, of this sequence is obtained as follows:

S = a₁ + a₂ + a₃ +.... +a_n-1 + a_n .... (1)

S = a₁ + a₂ +.... +a_n-2 + a_n-1 + a_n .... (2)

Subtracting (2) from (1), we get, a_n = a₁ + [(a₂ - a₁) + (a₃ - a₂) +.... + (a_n - a_{n - 1})].

Since the terms within the brackets are either in an A.P. or a G.P., we can find the value of a_n, the n^th term. We can now find the sum of the n terms of the sequence as

If corresponding to the sequence a₁, a₂, a₃,....,a_n, there exists a sequence b₀, b₁, b₂,...., b_n such that a_k = b_k - b_k-1, (k = 1, 2, 3, ....) then the sum of n terms of the sequence a₁, a₂,...a_n

is b_n - b₀.




Some Important Results

1. 1 + 2 + 3 +...+ n = (n + 1) (sum of first n natural numbers).



Inequalities

A.M. G. M. H. M. :

For two positive numbers a and b,

A.M. = , G. M. = , H.M. = so that

A.M. - G.M. =

and G.M. - H.M. =

=

A.M G.M. H.M.


Let a₁, a₂,...., a_n be n positive real numbers, then we define their arithmetic mean (A), geometric mean (G) and harmonic mean (H) as A =, G = (a₁ a₂ ... a_n)^1/n and H = .

It can be shown that A G H. Moreover equality holds at either place if and only if

a₁ = a₂ =.... = a_n.



Weighted Means:

Let a₁, a₂,.... , a_n be n positive real numbers and m₁, m₂,...., m_n be n positive rational numbers. Then we define weighted Arithmetic mean (A*), weighted Geometric mean (G*) and

weighted harmonic mean (H*) as

A* =, G*=and


H* =.


It can be shown that A* G* H*. Moreover equality holds at either place if and only

if a₁ = a₂ =....= a_n .