CHAPTER - 8 SEQUENCES AND SERIES KEY POINTS . In general, listing of any collection of objects in certain order is sequence. . A sequence is a function whose domain is the set N of natural numbers or some subset of it. e Let a1, a2, a3, .... ee be a sequence, then the expression a, + a2 + a3 + is called series associated with given sequence. . A sequence containing finite number of terms is called finite sequence. e A sequence is infinite, if it is not finite sequence. ° A sequence is said to be a progression if all the terms of the sequence can be expressed by same formula . Arithmetic Progression: A sequence is called an arithmetic progression if the difference between of a term and its previous term is always same, i.€., an+1— An = constant (=d) for all neN e General A.P. isa, a + d and a + 2d, where a = first term and d = common difference. . an =a+t(n—1)d =n" term of AP. =/ . S, = Sum of first n terms of A.P. = sla + J], where / = last term N. =F l2a+(n—1)a]If a, b, care in A.P. then a +k, b +k, c+kareinAP. ak, bk, ck also in A.P., k #0 are also in A.P. where k # 0. alo *|o xo If a, A, b are in A.P., then A is called arithmetic mean of a and b. Arithmetic mean between a and b is = 2 ; D If Ax, Az, As, ...-.An are n numbers inserted between a and b, such that the resulting sequence is A.P. then, A, = a+nd where d=2=2 n+1 Sk — Sit = ae In an A.P., the sum of the terms equidistant from the beginning and from the end is always same, and equal to the sum of the first and the last term. Ifa, b, care in A.P. then 2b=a+c. Three terms of A.P. can be chosen as a—d, a,at+d Four terms of A.P. can be chosen as a — 3d, a—d,a+d,a+ 3d. G.P. (Geometrical Progression) (i) avarar, wn (General G.P.) Where a = First term And r = common ratioIfa, b, care in G.P., then b? = ac. Ifa, G, b are in GP, then G is called geometric mean of a and b Geometric mean of two positive numbers a and b is Jab. If Gi, Go, Gs, ..........Gn are n numbers inserted between a and b so that the resulting sequence is G.P., then 1 (b\ni G, =ar"where r= (\"" a a Three terms of G.P. are chosen as —, a, ar. r aa Four terms of G.P. are chosen as 3, —,a, ar’. r If a, b, c are in G.P. then (i) are also in GP, (ii) ak, bk, ck als A|9 o]o o\/= are also in G.P., where k # 0 (iii) , care also in G.P. where k #0, a", b”, c” are also in GP. In a GP., the product of the terms equidistant from the beginning and from the end is always same and equal to the product of the first and the last term. Sum of infinite G.P. is possible if |r| < 1 and sum is given byMIND MAP Arrangement of number's ina definite order Sequence ‘Sequence AP ‘The difference between term and the term preceding to itis alway a constant. The constant called common difference = and series ay ay Itis the expression a)+a,+..+a, where Series a, isa sequence GP The difference between term and the term preceding to itis alway a constant. The constant term called common radio =r General term a+(n—1)d General form aatdat2d,a+3d, Ifa,b,¢ are in AP ‘Sum of n terms w/2f2a+(n—1)d) or Afata 2b=a+e Arthemetic Mean (A.M. ‘The number which when inserted between two numbers make the sequence A.P. Single AM between a, b aAM=2t (General form a,ar, ar’, ar’ General form sent Sum of n terms ifa, b, care in GP bi=ac ‘Geometric Mean (G.M.) The number which when inserted between two numbers make the 2 1AM. between a,b +nd A boa qa net Relation between AM. and G.M. AM2> GM. sequence GP. Single GM between a,b GM.= ab sleet ‘Sum of infante terms al nG.M. between a, b| G,=ar" isN a fF wo 10. 11. 12. 13. 14. 15. VERY SHORT ANSWER TYPE QUESTIONS Ifn™ term of an A.P. is 6n — 7 then write its 50" term. If Sq = 3n? + 2n, then write ag Which term of the sequence 3, 10, 17, ........... is 1367 fin an AP. 7 term is 9 and 9" term is 7, then find 16" term. If sum of first n terms of an A.P is 2n? + 7n, write its n" term. Which term of the G.P.2, 1, If in a G.P., a3 + as = 90 and if r = 2 find the first term of the G.P. In G.P.2V2, 4,.........128V2, find the 4" term from the end. If the product of 3 consecutive terms of G.P. is 27, find the middle term. Find the sum of first 8 terms of the G.P. 105.5, ceeeeeeeee Find the value of 5"? x 5" x 5" ....... upto infinity. Write the value of 0.3 [Hint: 0.3 = 0.3+.0.03 +0.003 +.. The first term of a G.P. is 2 and sum to infinity is 6, find common ratio. If 7" and 13" terms of an A.P. be 34 and 64 respectively, find 18" term Find geometric mean of 4 and 9.17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. Find If the sum of first p terms of an A.P. is q and sum of first q terms is p, then the sum of first p + q terms. Find sum to infinity of sequence 5, on ela If a, b, c are in A.P. and x, y, z are in G.P., then find the value of xP yr x 2°, Find two geometric means between numbers 1 and 64. 2n-3 Write third term of sequence whose general term is a, = SHORT ANSWER TYPE QUESTIONS Write the n" term of the series, > +5 _4_7 5 7AV 8.12% 9.13' Find the number of terms in the A.P. 7, 10, 13, ...... Inan A-P., 8, 11, 14, .......... find Sp - Sp_4 Find the sum of given terms:- (a) 81 + 82 + 83.0.2... + 89 +90 (b) 251 + 252 + 253 + ............ + 259 + 260 (a) Ifa, b, care in A.P. then show that 2b = atc. (b) Ifa, b, c are in G.P. then show that b? = a-c. If a, b, c are in G.P. then show that a” + b’, ab + be, b’ + c’are also in G.P. Find the least value of n for which 1+3+37+...+3""'> 100028 29. 30 31. 32. 33. 34. 35. 36. 37. 38. Write the first negative term of the sequence 20,194, 84, Determine the number of terms in A.P. 3, 7, 11, ........ 407. Also, find its 11" term from the end. How many numbers are there between 200 and 500, which leave remainder 7 when divided by 9. Find the sum of all the natural numbers between 1 and 200 which are neither divisible by 2 nor by 5. Find the sum of the sequence, T2470 + 68+ ccc... +40 ifin an A.p-22=2, find 24, ay 7 a, In an A.P. sum of first 4 terms is 56 and the sum of last 4 terms is 112. If the first term is 11 then find the number of terms. Solve: 1+6+11+16+.. +x = 148 The ratio of the sum of n terms of two A-P.'s is (7n — 1): (3n + 11), find the ratio of their 10" terms. If the I, 2" and last terms of an A.P are a, b and c respectively, then find the sum of all terms of the A.P. b+tc-2a cta-2b atb—2c If ,» ——— are in AP. then show that a a b sare also in A.P. [Hint. : Add 3 to each term] c oI o)o39. 40. 41. 42. 43. 44, 45. 46. 47. The product of first three terms of a G.P. is 1000. If 6 is added to its second term and 7 is added to its third term, the terms become in A.P. Find the G.P. If the continued product of three numbers in G.P. is 216 and the sum of their products in pairs is 156, find the numbers. Find the sum to infinity of the series: 4434547 2 2 2 IfA=1+1+/*+ ........... up to infinity, then express r in terms of ‘a’ and ‘A’. Find the sum of first terms of the series 0.7 + 0.77 + 0.777 + ..... iixeat2e 2 ; ysb-Pe eb. eo and ze ote oneea0 Prove that = 28. r Zz Cc The sum of first three terms of a G.P. is 15 and sum of next three terms is 120. Find the sum of first n terms. Prove thato.0037 = —. 225 [Hint: 0.031 = 0.03 + 0.001 + 0.0001 +..... Now use infinite G.P.] If a, b, c are in G.P. that the following are also in G.P. (i) a2, b?, c? (i) a3, b,c? (iii) Va, vb, VC are in GP.48. 49. 50. 51. 52. 53. 54 55. If a, b, c are in A.P. that the following are also in A.P: (ii) btc, cta, atb \ (ii ai 2), it +1) 4ih) are in AP. b c) ble al cla b If the numbers a”, b* and c” are given to be in A.P., show that 11 and — arein AP. bte’ cta +b zm . 353 Show that: 0.356 = —— 990 The n" term of a G.P. is 128 and the sum of its n term is 255. If its common ratio is 2, find the first term. The fourth term of a G.P. is 4. Find product of its first seven terms. If A, Az, As, Ag are four A.M’s between dana 3, then prove Ait Act Agt Ag = 7. If S, denotes the sum of first n terms of an A.P. If San = 5S, then 8.17 rove — = p 4 an LONG ANSWER TYPE QUESTIONS Prove that the sum of n numbers between a and b such that the resulting series becomes A.P. isan).56. 57. 58. 59. 60. 61. 62. 63. 1 If a, b, c are in G.P., then prove that>—y - >—z F-G- ae _ [Hint : Put b = ar, c = ar] Find two positive numbers whose difference is 12 and whose arithmetic mean exceeds the geometric mean by 2. If a is A.M. of b and c and c, G;, G2, b are in G.P., then prove that G? + G3 = abe The sum of an infinite G.P. is 57 and the sum of the cubes of its term is 9747, find the G.P. 37,15 Find the sum of first n terms of the series 14347418 2 4 8 16 terms. Hint: oat tasty... 2 4 8 16 Three positive numbers form an increasing G.P. If the middle term in the G.P. is doubled, then new numbers are in A.P. then find the common ratio of the G.P. Find three numbers in G.P. whose sum is 13 and the sum of whose squares is 91. The side of given space is 10 cm. The mid points at its, sides are joined to form a new square. Again the mid point of the sides of this new square are joined to form another square. This process is contined indefinitely. Based on the information answer the following questions.A P B (i) The side of first square is 10 cm what is the side of IInd square formed. (ii) What is the sum of area’s of all the square formed? (iii), What is the sum of perimeters of all the square formal? CASE STUDY TYPE QUESTIONS Abhishek buys Kisan Vikas Patra (KVP) from post office every year. Each year he exceeds the value of KVP by %1000 from last year's purchase. After 5 years he finds that the total value of KVP. purchased by him is 40,000.00. Based on the above information answer the following :- i. The sequence of amount of KVP forms a/an65. (a) Arithmetic Progression (b) Geometric Progression (c) Harmonic Progression (d) None of these ii. Find the amount of KVP purchased by him initially. (a) $7000 (b) %8000 (c) ¥6000 (d) %7500 iii. What will be the total amount of KVP purchased by him after 10 years? (a) €1,20,000 (b) 1,05,000 (c) €1,40,000 (d) €4,35,000 iv. What is the amount of KVP purchased by him in the 8" year? (a) 14,000 (b) 715,000 (c) 13,000 (a) 712,000 v. If he buys KVP every year for 10 years, how much will he spend in the purchase of last 4 KVP? (a) 765,000 (b) 754,000 (c) $75,000 (d) None of these A person writes a f letter to four of his friends. He asks » each one of them ee to copy the letter | ati and mail it to four different persons with the instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paisa to mail one letter, anwer the following questions.The sequence of letters mailed in each set forms a/an. (a) Arithmetic Progression (b) Geometric Progression (c) Harmonic Progression (d) None of these Find the number of letters mailed in the 4" set. (a) 64 (b) 16 (c) 256 (a) 1024 Find the total number of letters mailed in the first 5 sets. (a) 1364 (b) 1650 (c) 1236 (a) 1368 Find the amount spent on the postage when 8" set of letters is mailed? (a) %46,930 (b) 254,930 (c) 287,380 (d) 243,690 Find the amount spent on the mailing of 9" set? (a) %1,74,762 (b) %1,31,072 (c) %1,54,536 (d) None of these Multiple Choice Questions 66. 67. The interior angles of a polygon are in A.P. If the smallest angle be 120° and the common difference be 5, then the number of side is - (a) 8 (b) 10 (c) 9 (d) 6. a. and f are the roots of the equation x? - 3x + a = 0 and y and 8 are the roots of the equation x? — 12x + b = 0. If a, B, y and 5 form an increasing G.P., then (a, b)-68. 69. 70. 71. 72. (a) (3, 12) (b) (12, 3) (c) (2, 32) (d) (4, 16) If A be the arithmetic mean between two numbers and S be the sum of n arithmetic means between the same numbers, then - (a) S=nA (b) A=nS (c) A=S (d) None of these. If n geometric means be inserted between a and b, then the n" geometric mean will be- a mt (a) fer (b) f°) a a a 1 nel in (c) {| (d) 2) : a a If the arithmetic and geometric means of two numbers are 10 and 8 respectively, then one number exceeds the other number by- (a) 8 (b) 10 (c) 12 (d) 16 The first and last terms of A.P. are 1 and 11. If the sum of its term is 36, then the number of terms will be- (a) 5 (b) 6 (c) 7 (a) 8. If the first, second and last term of an A.P. are a, b and 2a respectively, then its sum is - ab (a) 2a) » 2 3ab (c) 2(b-a) (d) None of these.73. 74, 75. 76 77. If p", q” and r terms of an A.P. are in G.P., then the common ratio of this G.P.is - p-q q-r (a) (b) P-4 (c) par (d) None of these. If A be one A.M. and p, q be two GM's between two numbers, then 2A is equal to- pr+q? pg (a) Pa (b) Pd pe+g pq () 2 (d) 2. Ina GP. if the (m + n)"" term is p and (m —n)'" term is q, then its m" term is - (a) 0 (b) pq (o Pa @) d@+9). If S be the sum, P the product, R be the sum of reciprocals of n terms of G.P. then P? is equal to @ = &) = (3) (3) The n" term of a G.P. is 128 and the sum of its n terms is 225. If its common ratio is 2, then its first term is (a) 1 (b) 3 (c) 8 (d) none of these78. If second term of a GP. is 2 and the sum of its infinite term is 8, then its first term is 1 1 = b) — @ 5 ) 5 (c) 2 (d) 4 79. The two geometric means between the numbers 1 and 64 are (a) 1 and 64 (b) 4 and 16 (c) 2and 16 (d) 8 and 16 80. The product (32), (32)', (32)"° .... to 0 is equal to (a) 64 (b) 16 (c) 32 (a) 0 Directions: Each of these questions contains two statements, Assertion and Reason. Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select one of the codes (a), (b), (c) and (d) given below. Assertion is correct, reason is correct: reason is not a correct explanation for assertion. Assertion is correct, reason is correct: reason is not a correct explanation for assertion. (c) Assertion is correct , reason is incorrect. (d) Assertion is incorrect, reason is correct. (a (b} 81. Assertion: Value of a17. Whose n™ term is an = 4n — 3, is 65. Reason: Value of ag, whose nth term is an = (-1)"~*. n°. 82. Assertion: |f the third term of a G-P. is 4, then the product of its first five terms is 4°.83. 84, 85. 11. 13. 15. 17. Reason: Product of first five terms of a G.P. is given as a (ar) (ar’) (ar*) (ar'). Assertion: If a, b, c are in A.P. then b +c, c + a, a+b are in AP. Reason: If a, b, c are in A.P., then 10°, 10°, 10° are in G.P. . 2 3 31 Assertion: If —, k, — are in A.P., then the value of k is —. 3 8 48 Reason: Three numbers a, b, c are in AP. iff 2b=a+c Assertion: For x = + 1, the numbers are in G.P. Reason: Three numbers a, b, c are in G.P. if b’ = ac. 293 201 20" 4. 0 4an+5 6. 12th 3 8 64 (41 3 10. 20|1-= 2 5 12, 4 3 z 14. 89 6 16. -(p +a) 15/2 18. 119. 21. 23. 24, 28. 30 32. 34, 36. 39. 41. 43. 51. 57. 60. 62. 4 and 16 2n+1 (n+6)(n+10) 3n+5 855, 2555 a1 4 33 952 11 33:17 5, 10, 20, ......; or 20, 10, 5, 7 -n qilen-1+10 ] 1 16,4 n+27-1 1,3,9 20. 22. 27. 29. 31 33. 35. 37. 40. 42. 45. 52. 59. 61. 102, 367 7799 alo 36 (b+c—2a)(a+c) 2(b—a) 18, 6, 2; or 2, 6, 18 (“ay63. i. SV2om ii, 200 cm? ili. (80+ 402) om 64. i. (a) iit. (C) ili, (b) iv. (c) v. (b) 65. i. (b) ii. (c) ili, (a) iv. (d) v. (b) 66. (c) 67. (c) 68. (a) 69. (c) 70. (c) 71. (b) 72. (c) 73. (b) 74. (a) 75. (c) 76. (d) 77. (a) 78. (d) 79. (b) 80. (a) 81. (b) 82. (a) 83. (b) 84. (a) 85. (a)