Sequences and Series PDF

Chapter 8 SEQUENCES AND SERIES vNatural numbers are the product of human spirit. – DEDEKIND v 8.1 Introduction In mathematics, the word, “sequence” is used in much the same way as it is in ordinary English. When we say that a collection of objects is listed in a sequence, we usu...

Chapter 8 SEQUENCES AND SERIES vNatural numbers are the product of human spirit. – DEDEKIND v 8.1 Introduction In mathematics, the word, “sequence” is used in much the same way as it is in ordinary English. When we say that a collection of objects is listed in a sequence, we usually mean that the collection is ordered in such a way that it has an identified first member, second member, third member and so on. For example, population of human beings or bacteria at different times form a sequence. The amount of money deposited in a bank, over a number of years form a sequence. Depreciated values of certain commodity occur in a sequence. Sequences have important applications in several Fibonacci spheres of human activities. (1175-1250) Sequences, following specific patterns are called progressions. In previous class, we have studied about arithmetic progression (A.P). In this Chapter, besides discussing more about A.P.; arithmetic mean, geometric mean, relationship between A.M. and G.M., special series in forms of sum to n terms of consecutive natural numbers, sum to n terms of squares of natural numbers and sum to n terms of cubes of natural numbers will also be studied. 8.2 Sequences Let us consider the following examples: Assume that there is a generation gap of 30 years, we are asked to find the number of ancestors, i.e., parents, grandparents, great grandparents, etc. that a person might have over 300 years. Here, the total number of generations = 300 = 10 30 2024-25 136 MATHEMATICS The number of person’s ancestors for the first, second, third, …, tenth generations are 2, 4, 8, 16, 32, …, 1024. These numbers form what we call a sequence. Consider the successive quotients that we obtain in the division of 10 by 3 at different steps of division. In this process we get 3,3.3,3.33,3.333,... and so on. These quotients also form a sequence. The various numbers occurring in a sequence are called its terms. We denote the terms of a sequence by a1, a2, a3, …, an, …, etc., the subscripts denote the position of the term. The nth term is the number at the nth position of the sequence and is denoted by an. The nth term is also called the general term of the sequence. Thus, the terms of the sequence of person’s ancestors mentioned above are: a1 = 2, a2 = 4, a3 = 8, …, a10 = 1024. Similarly, in the example of successive quotients a1 = 3, a2 = 3.3, a3 = 3.33, …, a6 = 3.33333, etc. A sequence containing finite number of terms is called a finite sequence. For example, sequence of ancestors is a finite sequence since it contains 10 terms (a fixed number). A sequence is called infinite, if it is not a finite sequence. For example, the sequence of successive quotients mentioned above is an infinite sequence, infinite in the sense that it never ends. Often, it is possible to express the rule, which yields the various terms of a sequence in terms of algebraic formula. Consider for instance, the sequence of even natural numbers 2, 4, 6, … Here a1 = 2 = 2 × 1 a2 = 4 = 2 × 2 a3 = 6 = 2 × 3 a4 = 8 = 2 × 4................................................ a23 = 46 = 2 × 23, a24 = 48 = 2 × 24, and so on. In fact, we see that the nth term of this sequence can be written as an = 2n, where n is a natural number. Similarly, in the sequence of odd natural numbers 1,3,5, …, the nth term is given by the formula, an = 2n – 1, where n is a natural number. In some cases, an arrangement of numbers such as 1, 1, 2, 3, 5, 8,.. has no visible pattern, but the sequence is generated by the recurrence relation given by a1 = a2 = 1 a3 = a1 + a2 an = an – 2 + an – 1, n > 2 This sequence is called Fibonacci sequence. 2024-25  SEQUENCES AND SERIES 137 In the sequence of primes 2,3,5,7,…, we find that there is no formula for the nth prime. Such sequence can only be described by verbal description. In every sequence, we should not expect that its terms will necessarily be given by a specific formula. However, we expect a theoretical scheme or a rule for generating the terms a1, a2, a3,…,an,… in succession. In view of the above, a sequence can be regarded as a function whose domain is the set of natural numbers or some subset of it. Sometimes, we use the functional notation a(n) for an. 8.3 Series Let a1, a2, a3,…,an, be a given sequence. Then, the expression a1 + a2 + a3 +,…+ an +... is called the series associated with the given sequence.The series is finite or infinite according as the given sequence is finite or infinite. Series are often represented in compact form, called sigma notation, using the Greek letter ∑ (sigma) as means of indicating the summation involved. Thus, the series a1 + a2 + a3 +... + an is abbreviated n as ∑ ak. k =1 Remark When the series is used, it refers to the indicated sum not to the sum itself. For example, 1 + 3 + 5 + 7 is a finite series with four terms. When we use the phrase “sum of a series,” we will mean the number that results from adding the terms, the sum of the series is 16. We now consider some examples. Example 1 Write the first three terms in each of the following sequences defined by the following: n−3 (i) an = 2n + 5, (ii) an =. 4 Solution (i) Here an = 2n + 5 Substituting n = 1, 2, 3, we get a1 = 2(1) + 5 = 7, a2 = 9, a3 = 11 Therefore, the required terms are 7, 9 and 11. n−3 1− 3 1 1 (ii) Here an =. Thus, a1 = = − , a2 = − , a3 = 0 4 4 2 4 2024-25 138 MATHEMATICS Hence, the first three terms are – 1 , – 1 and 0. 2 4 Example 2 What is the 20th term of the sequence defined by an = (n – 1) (2 – n) (3 + n) ? Solution Putting n = 20 , we obtain a20 = (20 – 1) (2 – 20) (3 + 20) = 19 × (– 18) × (23) = – 7866. Example 3 Let the sequence an be defined as follows: a1 = 1, an = an – 1 + 2 for n ≥ 2. Find first five terms and write corresponding series. Solution We have a1 = 1, a2 = a1 + 2 = 1 + 2 = 3, a3 = a2 + 2 = 3 + 2 = 5, a4 = a3 + 2 = 5 + 2 = 7, a5 = a4 + 2 = 7 + 2 = 9. Hence, the first five terms of the sequence are 1,3,5,7 and 9. The corresponding series is 1 + 3 + 5 + 7 + 9 +... EXERCISE 8.1 Write the first five terms of each of the sequences in Exercises 1 to 6 whose nth terms are: n 1. an = n (n + 2) 2. an = 3. an = 2n n +1 2n − 3 n2 + 5 4. an = 5. an = (–1)n–1 5n+1 6. a n = n. 6 4 Find the indicated terms in each of the sequences in Exercises 7 to 10 whose nth terms are: n2 7. an = 4n – 3; a17, a24 8. an = 2n ; a7 n(n – 2) 9. an = (–1)n – 1n3; a9 10. an = ; a20. n+3 2024-25  SEQUENCES AND SERIES 139 Write the first five terms of each of the sequences in Exercises 11 to 13 and obtain the corresponding series: an −1 11. a1 = 3, an = 3an – 1 + 2 for all n > 1 12. a1 = – 1, an = ,n≥2 n 13. a1 = a2 = 2, an = an – 1–1, n > 2 14. The Fibonacci sequence is defined by 1 = a1 = a2 and an = an – 1 + an – 2, n > 2. an +1 Find an , for n = 1, 2, 3, 4, 5 8.4 Geometric Progression (G. P.) Let us consider the following sequences: 1 –1 1 –1 (i) 2,4,8,16,..., (ii) , , , (iii).01,.0001,.000001,... 9 27 81 243... In each of these sequences, how their terms progress? We note that each term, except the first progresses in a definite order. a2 a a In (i), we have a1 = 2, = 2, 3 = 2, 4 = 2 and so on. a1 a2 a3 1 a 1 a 1 a 1 In (ii), we observe, a1 = , 2 = , 3 = , 4 = and so on. 9 a1 3 a2 3 a3 3 Similarly, state how do the terms in (iii) progress? It is observed that in each case, every term except the first term bears a constant ratio to the term immediately preceding 1 it. In (i), this constant ratio is 2; in (ii), it is – 3 and in (iii), the constant ratio is 0.01. Such sequences are called geometric sequence or geometric progression abbreviated as G.P. A sequence a1, a2, a3, …, an, … is called geometric progression, if each term is ak + 1 non-zero and = r (constant), for k ≥ 1. ak By letting a1 = a, we obtain a geometric progression, a, ar, ar2, ar3,…., where a is called the first term and r is called the common ratio of the G.P. Common ratio in 1 geometric progression (i), (ii) and (iii) above are 2, – and 0.01, respectively. 3 2024-25 140 MATHEMATICS As in case of arithmetic progression, the problem of finding the nth term or sum of n terms of a geometric progression containing a large number of terms would be difficult without the use of the formulae which we shall develop in the next Section. We shall use the following notations with these formulae: a = the first term, r = the common ratio, l = the last term, n = the numbers of terms, Sn = the sum of first n terms. 8.4.1 General term of a G.P. Let us consider a G.P. with first non-zero term ‘a’ and common ratio ‘r’. Write a few terms of it. The second term is obtained by multiplying a by r, thus a2 = ar. Similarly, third term is obtained by multiplying a2 by r. Thus, a3 = a2r = ar2, and so on. We write below these and few more terms. 1 term = a1 = a = ar1–1, 2nd term = a2 = ar = ar2–1, 3rd term = a3 = ar2 = ar3–1 st 4th term = a4 = ar3 = ar4–1, 5th term = a5 = ar4 = ar5–1 Do you see a pattern? What will be 16th term? a16 = ar16–1 = ar15 Therefore, the pattern suggests that the nth term of a G.P. is given by an = ar n–1. Thus, a, G.P. can be written as a, ar, ar2, ar3, … arn – 1; a, ar, ar2,...,arn – 1... ;according as G.P. is finite or infinite, respectively. The series a + ar + ar2 +... + arn–1 or a + ar + ar2 +... + arn–1 +...are called finite or infinite geometric series, respectively. 8.4.2. Sum to n terms of a G.P. Let the first term of a G.P. be a and the common ratio be r. Let us denote by Sn the sum to first n terms of G.P. Then Sn = a + ar + ar2 +...+ arn–1... (1) Case 1 If r = 1, we have Sn = a + a + a +... + a (n terms) = na Case 2 If r ≠ 1, multiplying (1) by r, we have rSn = ar + ar2 + ar3 +... + arn... (2) Subtracting (2) from (1), we get (1 – r) Sn = a – arn = a(1 – rn) a( r n −1) This gives or Sn = r −1 Example 4 Find the 10th and nth terms of the G.P. 5, 25,125,…. Solution Here a = 5 and r = 5. Thus, a10 = 5(5)10–1 = 5(5)9 = 510 and an = arn–1 = 5(5)n–1 = 5n. 2024-25  SEQUENCES AND SERIES 141 Example 5 Which term of the G.P., 2,8,32,... up to n terms is 131072? Solution Let 131072 be the nth term of the given G.P. Here a = 2 and r = 4. Therefore 131072 = an = 2(4)n – 1 or 65536 = 4n – 1 This gives 48 = 4n – 1. So that n – 1 = 8, i.e., n = 9. Hence, 131072 is the 9th term of the G.P. Example 6 In a G.P., the 3rd term is 24 and the 6th term is 192.Find the 10th term. Solution Here, a3 = ar 2 = 24... (1) and a6 = ar = 192 5... (2) Dividing (2) by (1), we get r = 2. Substituting r = 2 in (1), we get a = 6. Hence a10 = 6 (2)9 = 3072. Example 7 Find the sum of first n terms and the sum of first 5 terms of the geometric 2 4 series 1 + + +... 3 9 2 Solution Here a = 1 and r =. Therefore 3   2 n  1 −     n  a (1 − r n )   3   3 1−  2   Sn = = =   1− r 2   3   1− 3   2 5  211 211 In particular, S5 = 3 1 −    = 3 × =.   3   243 81 3 3 Example 8 How many terms of the G.P. 3, , ,... are needed to give the 2 4 3069 sum 512 ? 1 3069 Solution Let n be the number of terms needed. Given that a = 3, r = and Sn = 2 512 a (1 – r n ) Since Sn = 1− r 2024-25 142 MATHEMATICS 1 3(1 − ) Therefore 3069 = 2n = 6  1− 1  1  n  512 1−  2  2 3069 1 or = 1− n 3072 2 1 3069 3 1 or = 1− = = 2n 3072 3072 1024 or 2n = 1024 = 210, which gives n = 10. 13 Example 9 The sum of first three terms of a G.P. is and their product is – 1. 12 Find the common ratio and the terms. a Solution Let , a, ar be the first three terms of the G.P. Then r a 13 + ar + a =... (1) r 12 a... (2) and   (a ) (ar ) = – 1 r From (2), we get a3 = – 1, i.e., a = – 1 (considering only real roots) Substituting a = –1 in (1), we have 1 13 – –1 – r = or 12r2 + 25r + 12 = 0. r 12 3 4 This is a quadratic in r, solving, we get r = –or –. 4 3 4 3 –3 3 4 –4 Thus, the three terms of G.P. are : , – 1, for r = and , – 1, for r = , 3 4 4 4 3 3 Example10 Find the sum of the sequence 7, 77, 777, 7777,... to n terms. Solution This is not a G.P., however, we can relate it to a G.P. by writing the terms as Sn = 7 + 77 + 777 + 7777 +... to n terms 2024-25  SEQUENCES AND SERIES 143 7 = [9 + 99 + 999 + 9999 +...to n term] 9 7 = [(10 − 1) + (102 − 1) + (103 −1) + (104 − 1) +...n terms] 9 7 = [(10 + 102 + 103 +...n terms) – (1+1+1+...n terms)] 9 7  10(10n − 1)  7  10 (10n − 1)  =  − n  =  − n. 9  10 − 1  9 9  Example 11 A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own. Solution Here a = 2, r = 2 and n = 10 a (r n −1) Using the sum formula Sn = r −1 We have S10 = 2(210 – 1) = 2046 Hence, the number of ancestors preceding the person is 2046. 8.4.3 Geometric Mean (G.M.) The geometric mean of two positive numbers a and b is the number ab. Therefore, the geometric mean of 2 and 8 is 4. We observe that the three numbers 2,4,8 are consecutive terms of a G.P. This leads to a generalisation of the concept of geometric means of two numbers. Given any two positive numbers a and b, we can insert as many numbers as we like between them to make the resulting sequence in a G.P. Let G1, G2,…, Gn be n numbers between positive numbers a and b such that a,G1,G2,G3,…,Gn,b is a G.P. Thus, b being the (n + 2)th term,we have 1 n +1  b n + 1. b = ar , or r=  a 1  b  n+1 2 3 Hence G1 = ar = a   ,  b  n+1  b  n +1 a G 2 = ar = a   , G 3 = ar = a   , 2 3 a a n  b  n +1 G n = ar n = a   a 2024-25 144 MATHEMATICS Example12 Insert three numbers between 1 and 256 so that the resulting sequence is a G.P. Solution Let G1, G2,G3 be three numbers between 1 and 256 such that 1, G1,G2,G3 ,256 is a G.P. Therefore 256 = r4 giving r = ± 4 (Taking real roots only) For r = 4, we have G1 = ar = 4, G2 = ar2 = 16, G3 = ar3 = 64 Similarly, for r = – 4, numbers are – 4,16 and – 64. Hence, we can insert 4, 16, 64 between 1 and 256 so that the resulting sequences are in G.P. 8.5 Relationship Between A.M. and G.M. Let A and G be A.M. and G.M. of two given positive real numbers a and b, respectively. Then a+b A= and G = ab 2 Thus, we have a +b a + b − 2 ab A–G= − ab = 2 2 ( ) 2 a− b = ≥0... (1) 2 From (1), we obtain the relationship A ≥ G. Example 13 If A.M. and G.M. of two positive numbers a and b are 10 and 8, respectively, find the numbers. a+b Solution Given that A.M. = =10... (1) 2 and G.M. = ab = 8... (2) From (1) and (2), we get a + b = 20... (3) ab = 64... (4) Putting the value of a and b from (3), (4) in the identity (a – b) = (a + b)2 – 4ab, 2 we get (a – b)2 = 400 – 256 = 144 or a – b = ± 12... (5) 2024-25  SEQUENCES AND SERIES 145 Solving (3) and (5), we obtain a = 4, b = 16 or a = 16, b = 4 Thus, the numbers a and b are 4, 16 or 16, 4 respectively. EXERCISE 8.2 5 5 5 1. Find the 20th and nth terms of the G.P. , , ,... 2 4 8 2. Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2. 3. The 5 th, 8 th and 11 th terms of a G.P. are p, q and s, respectively. Show that q2 = ps. 4. The 4th term of a G.P. is square of its second term, and the first term is – 3. Determine its 7th term. 5. Which term of the following sequences: (a) 2, 2 2 , 4,... is 128 ? (b) 3 ,3,3 3 ,... is729 ? 1 1 1 1 (c) , , ,... is ? 3 9 27 19683 2 7 6. For what values of x, the numbers – , x, – are in G.P.? 7 2 Find the sum to indicated number of terms in each of the geometric progressions in Exercises 7 to 10: 7. 0.15, 0.015, 0.0015,... 20 terms. 8. 7 , 21 , 3 7 ,... n terms. 9. 1, – a, a2, – a3,... n terms (if a ≠ – 1). 10. x3, x5, x7,... n terms (if x ≠ ± 1). 11 11. Evaluate ∑ (2 + 3k ). k =1 39 12. The sum of first three terms of a G.P. is and their product is 1. Find the 10 common ratio and the terms. 13. How many terms of G.P. 3, 32, 33, … are needed to give the sum 120? 14. The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P. 15. Given a G.P. with a = 729 and 7th term 64, determine S7. 2024-25 146 MATHEMATICS 16. Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term. 17. If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P. 18. Find the sum to n terms of the sequence, 8, 88, 888, 8888…. 19. Find the sum of the products of the corresponding terms of the sequences 2, 4, 8, 1 16, 32 and 128, 32, 8, 2,. 2 20. Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … ARn – 1 form a G.P, and find the common ratio. 21. Find four numbers forming a geometric progression in which the third term is greater than the first term by 9, and the second term is greater than the 4th by 18. 22. If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that aq – r br – pcP – q = 1. 23. If the first and the nth term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that P2 = (ab)n. 24. Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from 1 (n + 1)th to (2n)th term is. rn 25. If a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2. 26. Insert two numbers between 3 and 81 so that the resulting sequence is G.P. a n+1 + b n+1 27. Find the value of n so that may be the geometric mean between a n + bn a and b. 28. The sum of two numbers is 6 times their geometric mean, show that numbers ( )( are in the ratio 3 + 2 2 : 3 − 2 2. ) 29. If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are A ± ( A + G )( A − G ). 30. The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2nd hour, 4th hour and nth hour ? 2024-25  SEQUENCES AND SERIES 147 31. What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually? 32. If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation. Miscellaneous Examples Example 14 If a, b, c, d and p are different real numbers such that (a2 + b2 + c2)p2 – 2(ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P. Solution Given that (a2 + b2 + c2) p2 – 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0... (1) But L.H.S. = (a2p2 – 2abp + b2) + (b2p2 – 2bcp + c2) + (c2p2 – 2cdp + d2), which gives (ap – b)2 + (bp – c)2 + (cp – d)2 ≥ 0... (2) Since the sum of squares of real numbers is non negative, therefore, from (1) and (2), we have, (ap – b)2 + (bp – c)2 + (cp – d)2 = 0 or ap – b = 0, bp – c = 0, cp – d = 0 b c d This implies that = = =p a b c Hence a, b, c and d are in G.P. Miscellaneous Exercise On Chapter 8 1. If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that n f(1) = 3 and ∑ f (x) = 120 , find the value of n. x =1 2. The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms. 3. The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P. 4. The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers. 5. A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio. 2024-25 148 MATHEMATICS a + bx b + cx c + dx 6. If = = ( x ≠ 0) , then show that a, b, c and d are in G.P. a − bx b − cx c − dx 7. Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn. 8. If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P. 9. If a and b are the roots of x2 – 3x + p = 0 and c, d are roots of x2 – 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q – p) = 17:15. 10. The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show ( )( that a : b = m + m2 – n 2 : m – m2 – n 2. ) 11. Find the sum of the following series up to n terms: (i) 5 + 55 +555 + … (ii).6 +. 66 +. 666+… 12. Find the 20th term of the series 2 × 4 + 4 × 6 + 6 × 8 +... + n terms. 13. A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How much will the tractor cost him? 14. Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him? 15. A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8th set of letter is mailed. 16. A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years. 17. A manufacturer reckons that the value of a machine, which costs him Rs. 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years. 18. 150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed. 2024-25  SEQUENCES AND SERIES 149 Summary ® By a sequence, we mean an arrangement of number in definite order according to some rule. Also, we define a sequence as a function whose domain is the set of natural numbers or some subsets of the type {1, 2, 3,....k}. A sequence containing a finite number of terms is called a finite sequence. A sequence is called infinite if it is not a finite sequence. ® Let a1, a2, a3,... be the sequence, then the sum expressed as a1 + a2 + a3 +... is called series. A series is called finite series if it has got finite number of terms. ® A sequence is said to be a geometric progression or G.P., if the ratio of any term to its preceding term is same throughout. This constant factor is called the common ratio. Usually, we denote the first term of a G.P. by a and its common ratio by r. The general or the nth term of G.P. is given by an= arn – 1. The sum Sn of the first n terms of G.P. is given by Sn = ( a rn – 1 ) or a (1– r ) , if r ≠ 1 n r –1 1–r ® The geometric mean (G.M.) of any two positive numbers a and b is given by ab i.e., the sequence a, G, b is G.P. Historical Note Evidence is found that Babylonians, some 4000 years ago, knew of arithmetic and geometric sequences. According to Boethius (510), arithmetic and geometric sequences were known to early Greek writers. Among the Indian mathematician, Aryabhatta (476) was the first to give the formula for the sum of squares and cubes of natural numbers in his famous work Aryabhatiyam, written around 499. He also gave the formula for finding the sum to n terms of an arithmetic sequence starting with p th term. Noted Indian mathematicians Brahmgupta 2024-25 150 MATHEMATICS (598), Mahavira (850) and Bhaskara (1114-1185) also considered the sum of squares and cubes. Another specific type of sequence having important applications in mathematics, called Fibonacci sequence, was discovered by Italian mathematician Leonardo Fibonacci (1170-1250). Seventeenth century witnessed the classification of series into specific forms. In 1671 James Gregory used the term infinite series in connection with infinite sequence. It was only through the rigorous development of algebraic and set theoretic tools that the concepts related to sequence and series could be formulated suitably. —v — 2024-25