Sequences and Series Examples

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• (i) Let X be a set of numbers and  f : N_n → 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 {T_n} is a sequence, then the sum T₁ +T₂ + T₃.... is called the series corresponding to the sequence {T_n}. 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 n^th 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 {T_n} is said to be an arithmetic progression (A.P) if there exists a number, say d such  that     T_n+1 - T_n = 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. {T_n}, then the sum of first n terms, S_n 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 T_n = S_n - S_n-1.

(v) If the sequence a, A₁, A₂,.....,A_n, b is an A.P., then the numbers A₁, A₂,.....,A_n 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 {T_n} 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. {T_n}, then the sum of first n terms, S_n 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,....,G_n, b of positive numbers is a G.P., then the numbers G₁, G₂,....,G_n, 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 n^th 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,.....,H_n, b of non-zero numbers is a H.P., then the numbers H₁, H₂,.....,H_n 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