Sequence and Series Exercise

SEQUENCE AND SERIES EXERCISE Q.1 Q.2 Q.3 Write first 4 terms in each of the sequences: (2n  3) (i) an = 5n + 2 (ii) an = 2 (iii) an = (1)n  1 × 2n + 1 Find first five terms of the sequence, defined by a1 = 1, an = an  1 + 3 for n  2 Find first 5 terms of the sequence, defined by a n1 a1 = 1, an = for n  2. n Q.4 Q.5 Find 23rd term of the AP 7, 5, 3, 1, ….. . Q.19 Q.20 Q.21 The angles of a quadrilateral are in AP whose common difference is 10o. Find the angles. Q.22 Divide 32 into four parts which are in AP such that the product of the extremes is to the product of means as 7 : 15. Q.23 Find the sum of 24 terms of the AP 1, 3, 5, 7, … . Q.24 Find the sum of 10 terms of the AP Find the 20 term of the AP th 2 ,3 2 ,5 2 , …… . Three numbers are in AP. If their sum is 27and the product 648, find the numbers. The sum of three consecutive terms of an AP is 21 and the sum of the squares of these terms is 165. Find these terms. 1 2 6,5 , 4 ,..... . 3 3 Q.6 Find the nth term of the AP 8, 3, 2, 7, …. . Q.7 Which term of the AP 5, 8, 11, 14, …Is 320? Q.8 Which term of the AP 64, 60, 56, ….. is 0? Q.9 How many terms are there in the AP 10, 13, 16, …. , 43 ? Q.26 Find the sum of 100 terms of AP 0.7, 0.71, 0.72, …... Q.10 How Q.27 Find the sum of the series 101 + 99 + 97 + ….+ 47. Q.28 How many terms of the AP 26, 21, 16, 11, … are needed to give the sum 11? Q.29 How many terms of the AP 18, 16, 14, 12,….are needed to give the sum 78? Explain the double answer. Q.30 Find the sum of 32 terms of an AP whose third term is 1 and the 6th term is 11. Q.31 In an AP, the first term is 2 and the sum of first is one-fourth of the next five terms. Show that its 20th term is 112. Q.32 If the sum of certain number of terms of the AP 25, 22, 19, …. Is 115, find the last term. Q.33 The sum of n term of two arithmetic AP many terms are there in the 5 1 1 ,1,1 ......,3 ? 6 6, 3 Q.11 Is 302 a term of the AP 3, 8, 13, 18, …. ? Q.12 The 5th and 13th terms of an AP are 5 and 3 respectively. Find this AP and obtain its 16th term. Q.13 The 2nd,31st and the last terms of an A.P. are 3 1 1 7 , and 6 respectively. Find the first term. 4 2 2 Q.14 If the 9th term of an A.P. is 0, prove that its 29th term is double the 19th term. Q.15 How many two-digit numbers are divisible by 7? Q.16 The 4th term of an AP is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference. Q.17 Q.18 Find the 15th term from the end of the AP 3, 5, 7, 9,….., 201. If 7 times the 7th term of an AP is equal to 11 times its 11th term, show that its 18th term is 0. Q.25 Find the sum of 20 terms of the AP 2, 2 2,3 2 ,…… progressions are in the ratio (3n + 8) : (7n + 15). Find the ratio of their 12th terms. Q.34 The sum of n term of two arithmetic progressions are in the ratio (7n  5) : (5n + 17). Show that their 6th terms are equal. Q.35 If the m th term of an AP is a and its n th term is b, show that the sum of its (m + n) terms is SEQUENCE AND SERIES (n  m)  (a  b)  a  b   2 ( m  n)  .  Q.36 If S1, S2, S3, …., Sm are the sums of n terms of m APs whose first terms are 1, 2, 3, …., m and common differences are 1, 3, 5, …., (2m 1) respectively, show that (S1 + S2 + S3 + …..+ Sm) = 1 mn(mn + 1). 2 Q.37 The sum of first 7 terms of an AP is 10 and that of next 7 terms is 17. Find the AP. Q.38 If the sum of n terms of an AP is given by Sn = (3n2 + 4n), find its rth term. Q.39 If the sum of n terms of an AP is (3n2 + 5n) and its mth term is 164, find the value of m. Q.40 Arun buys a scooter for Rs.22000. He pays Rs 4000 in cash and agrees to pay the balance in annual instalments of Rs l000 each plus 10% interest on the unpaid amount. How much will the scooter cost him? Q.41 Q.42 Q.43 Q.44 Q.45 Q.46 Q.47 speed by 1km each succeeding hour. After how many hours will the second car overtake the first car if both cars go non-stop? There are 30 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardner waters all the trees separately starting from the well and return to the well after watering each trees to get water for the next. Find the total distance the gardner will cover in order to water all the trees. Q.48 A sum of Rs 62400 is paid off in 30 instalments such that each instalment is Rs 100 more than the preceding instalment. Calculate the first instalment. Q.49 A circle is completely divided into n sectors in such a way that the angles of the sectors are in AP. If the smallest of these angles is 8o and the largest is 72o, calculate n and the angle in the fifth sector. Q.50 Ashok buys an old scooter for Rs l2000. He pays Rs 6000 in cash and agrees to pay the balance in annual instalments of Rs 500 each plus 12% interest on the unpaid amount. How much will the scooter cost him? The interior angles of a polygon are in AP. The smallest angle is 52o and the common difference is 8o. Find the number of sides of the polygon. Q.51 A man repays a loan of Rs 3250 by paying Rs 20 in the first month and then increasing it by Rs 15 every month. How long will it take him to clear the loan? The digits of a three-digit number are in AP and their sum is 15. The number obtained by reversing the digits is 594 less then the original number. Find the number. Q.52 Find AM between : 9 and 19 Q.53 Insert three numbers between 3 and 19 such that the resulting sequence is an AP. Q.54 Insert four numbers between 4 and 29 such that the resulting sequence is an AP. Q.55 Insert five numbers between 8 and 26 such that the resulting sequence is an AP. Q.56 Between 1 and 31. m numbers have been inserted in such a way that the ratio of 7th and A man saved Rs 16500 in 10 years. In each year after the first year he saved Rs 100 more than he did in the preceding year. How much did he save in the first year? 150 workers were engaged to finish a piece of work in a certain number of days. Four workers dropped out on second day, four more workers dropped out on third day, and so on. It takes 8 more days to finish the work now. Find the number of days in which the work was completed. A manufacturer of TV sets produced 6000 units in the third year and 7000 units in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find the production (i) in the first year,(ii)in the l0 year (iii) in 7 years Two cars start together in the same direction from the same place. The first goes with uniform speed of 40 km/hr. The second goes at a speed of 36 km/hr in the first hour and increases the (m  1)th numbers is 5 : 9. Find the value of m. Q.57 If a, b, c are in AP, prove that: (a  c)2 = 4(a  b)(b  c) Q.58 If a, b, c are in AP, show that (b + c (c + a  b), (a + b  c) are in AP. Q.59 If a, b, c are in AP, show that (a+2bc)(2b+ca)(c+ab) = 4abc. Q.60 If a2, b2, c2 are in AP, show that a b c , , are in AP.. b c c a ab  a), SEQUENCE AND SERIES Q.61 If b c c a ab , , are in AP, prove that a b c 1 1 1 , , are in AP.. a b c 1 1 1 Q.62 If , , are in AP, prove that a b c Q.63 Q.64 Find the 6th term from the end of the GP 8, 4, 2, …., Q.80 1 . 1024 Find the 4th term from the end of the GP 2 2 2 , , ,....,162 . 27 9 3 b c c a ab , , are in AP.. a b c Q.81 If a2(b + c), b2(c + a), c2(a + b) are in AP, show that either a, b, c are in AP or (ab + bc + ca) = 0. Find the sum of the GP 1 + 3 + 9 + 27 +…to 12 terms Q.82 Find the sum of the GP Find the 10th and nth terms of the GP Q.83 12, 4, Q.65 Q.79 4 4 , ,….. . 3 9 2 ,4, 8 2 , …. . Q.66 Find the 7th and nth terms of the GP 0.4, 0.8, 1.6, ….. . Q.67 Find the 10 th The first term of a GP is 27 and its 8th term is 1 . Find the sum of its first 10 terms. 81 1 1 Q.84 The 2nd and 5th terms of a GP are and 2 16 Find the 17th and nth terms of the GP 2, 2 7  21  3 7 +… to n terms respectively. Find the sum of the GP up to 8 terms. and nth terms of the GP Q.85 3 1 1 2 , , , , ….. . 4 2 3 9 The 4th and 7th terms of a GP are 1 and 27 1 respectively. Find the sum of n terms of 729 Q.68 Which term of the GP 3, 6, 12, 24, … is 3072? Q.69 Which term of the GP Q.70 Which term of the GP the GP. Q.86 The common ratio of a finite GP is 3 and its last term is 486. If the sum of these terms is 728, find the first term. 3 , 3, 3 3 , … is 729? Q.87 Q.71 Find the geometric series whose 5th and 8th terms are 80 and 640 respectively. Find the sum of the geometric series 3 + 6 + 12 + … + 1536. Q.88 How many terms of the series 2 + 6 + 18 + …. Must be taken to make the sum equal to 728? Q.72 Find the GP whose 4th and 7th terms are Q.89 How many terms of the geometric series 1 1 , ,1 ,….. is 128? 4 2 1 and respectively.. 486 Q.73 For what values of x are the numbers in GP? Q.90 The 5th, 8th, and 11th, terms of a GP are a, b, c respectively. Show that b2 = ac. The sum of n terms of a progression is (2n 1). Show that it is a GP and find its common ratio. Q.91 The first term of a GP is 3 and the square of the second term is equal to its 4th term. Find its 7th term. In a GP, the ratio of the sum of first 3 terms is to that of first 6 terms is 125 : 152. Find the common ratio. Q.92 Find the sum of the series 8 + 88 + 888 + … to n terms. Q.93 Find the sum of the series 3 + 33 + 333 + … to n terms. Q.94 Find the sum of the series 0.7 + 0.77 + 0.777 + … to n terms. For what values of x are the numbers (x + 9), (x  6) and 4 in GP? Q.75 Q.76 The sum of three numbers in GP is product is 1. Find the numbers. Q.78 2 1 1    .... must be taken to make the sum 9 3 2 55 equal to ? 72 2 7 , x, 7 2 Q.74 Q.77 1 18 39 and their 10 The sum of three numbers in GP is 21 and the sum of their squares is 189. Find the numbers. SEQUENCE AND SERIES 11 k Q.95 Evaluate  (2  3 ) k 1 Q.96 Q.97 Q.98 Q.99 The inventor of the chessboard suggested a reward of one grain of wheat for the first square; 2 grains for the second; 4 grains for the third, and so on doubling the number of grains for subsequent squares. How many grains would have to be given to the inventor? A man 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 the instruction that they move the chain similarly. Assuming that the chain is not broken and it costs Rs 2 to mail one letter, find the amount spent on postage when 8th set of letters is mailed. A manufacturer reckons that the value of a machine which costs him Rs 31250, will depreciate each year by 20%. Find the estimated value at the end of 5 years. Find two positive numbers a and b whose AM and GM are 25 and 20 respectively. Q.100 Find two positive numbers a and b whose AM and GM are 10 and 8 respectively. Q.101 Find the GM between 5 and 125 Q.102 Insert two numbers between 9 and 243 so that the resulting sequence is a GP. 1 Q.103 Inset three numbers between and 432 so 3 that the resulting sequence is a GP. Q.104 Insert four numbers between 6 and 192 so that the resulting sequence is a GP. Q.105 The AM between two positive numbers a and b (a > b) is twice their GM. Prove that a : b = (2 + 3) : ( 2  3 ). Q.106 If a, b, c are in AP, x is the GM between a and b; y is the GM between b and c; then show that b2 is the AM between x2 and y2. Q.107 Show that the product of n geometric means between a and b is equal to the nth power of the single GM between a and b. Q.108 If a, b, c are in GP, prove that 1 1 1 , , (a  b) 2b (b  c) are in AP.. Q.109 If a, b, c, d are in GP, prove that (b + c)(b + d) = (c + a)(c + d) Q.110 If a, b, c, d are in GP, prove that (i) (a + b), (b + c), (c + d) are in GP. Q.111 If a, b, c, d are in GP, show that 1 1 1 , , are in GP.. (a2  b2 ) (b2  c2 ) (c2  d2 ) 2 2 2 Q.112 If (p + q ), (pq + qr), (q + r2) are in GP, prove that p, q, r are in GP. Q.113 If a, b, c are in AP, and a, x, b and b, y, c are in GP then show that x2, b2, y2 are in AP. Q.114 If a, b, c are in AP and a, b, d are in GP, show that a, (a  b) and (d  c) are in GP. Q.115 Find the sum of the series whose nth is given by: (3n2 + 2n) Q.116 Find the sum of the series: (22 + 42 + 62 + 82 + …. To n terms) Q.117 Find the sum of the series: (1 × 2 × 4) + (2 × 3 × 7) + (3 × 4 × 10) + ….to n terms Q.118 Find the sum of the series: 1 1 1   + ….. to n terms (1 2) (2  3) (3  4) Q.119 Find the sum of each of the following infinite 2 + 4 + 2 2 + ….  series: 8 + 4 Q.120 Prove that 1 3 1 9 1 27 9  9  9  ..... = 3. Q.121 Using geometric series, express rational number. 0.3 as a Q.122 Using geometric series express 0.231 as a rational number. Q.123 Find the value of 0.423 in the form of a simple fraction. Q.124 Find the rational number whose decimal from is 0.356 . Q.125 Find the value of fraction. 3.52 in the form of a simple Q.126 The sum of an infinite geometric series is 6. If its first term is 2, find its common ratio. Q.127 One side of an equilateral triangle is 24 cm. The midpoints of its sides are joined to form another triangle whose midpoints, in turn, are joined to form still another triangle. The process continues indefinitely. Find the sum of the perimeters of all the triangles. Q.128 An equilateral is drawn by joining the midpoints of the sides of a given equilateral triangle. A third equilateral triangle is drawn inside the second triangle in the same manner. This process is repeated indefinitely. If each side of the first equilateral triangle is 6cm, find the sum of the areas of all the triangles. SEQUENCE AND SERIES ANSWER KEY 1. (i) 7, 12, 17, 22 (ii) 2. 1, 4, 7, 10, 13 3. –1, 1 1 1 1 , , , 2 6 24 120  5n) 7. 106th 8. 17th 9. 12 12. 9, 8, 7, 6, … and a16 = 6 13. a = 8 & n = 59 15. 13 16. a = 3, d = 2 17. 173 19. 6, 9, 12 20. 4, 7, 10 21. 75°, 85°, 95°, 105° 22. 2, 6, 10, 14 23. 576 210 2 26. 119.5 29. 6, 13 2 33. 7 : 16 38. 6r + 1 39. m = 27 4. 37 1 1 3 5 , , , (iii) 4,–8,16,–32 4 4 4 4 5. 39 2 6. (13 27. 2072 37. 28. 11 10. 16 30. 1696 11. No 24. 30 2 5 . 31. x = –2 32. x = 1 2 1  1  1  ..... 7 7 40. Rs 39100 41. Rs 16680 42. 20 months43. Rs 1200 44. 25 days 45. 5500 units (ii) 7750 units (iii) 43750 units 46. 9 hours 47. 4795 m 48. Rs. 630 49. n = 9, 5th angle = 40o 50. Three 51. 852 52. 14 53. 7, 11, 15 54. 9, 14, 19, 24 55. 11, 14, 17, 20, 13 56. m = 14 64. 68. 11th 69. 10th 70. 12th 71. 5 + 10 + 20 + 40 +….. 74. 0 or 16 76. –2187 2 5 5 2 77.  , 1 ,  or  , 1 ,  78. (3, 6, 12) or (12, 6, 3) 2   5 5 2 80. 6 81. 265720 82. 86. 2 87. 3069 93. 1 .(10n 1  9n  10) 27 7 2  n 4 4 128 3  2  1 , n 2 65. 512, 2 ( n1) 66. 25.6, 2 67. ,   2 5 6561 3 6561 4  3    3  1 3n / 2  1 88. 6 89. 5 7  1  94. 81  9n  1   10n  83. 72. 3 1 1 , , 2 2 6 81  1  . 1  10  84. 85 128 2  3  3 5 90. 2 91. 92. 95. 265741 96. (264  1) 73. 1 or 1 1 32 79. 85. 3  1 . 1  n  2  3    8 10n1  10  9n 81 97. Rs. 131072 98. Rs. 10240 99. (a = 40, b = 10) or (a = 10, b = 40) 100. (a = 4, b = 16) or (a = 16, b = 4) 102. 27, 81 103. (2, 12, 72) or (2, 12, 72) 101. 25 115. 1 n( n  1)(2n  3) 2   119. 8 2  2 121. 127. 144 cm 1 3 128. 12 116. 122. 3 cm2 1 2 n( n  1)(2n  1) 3 231 999 123. 419 990 104. 12, 24, 48, 96 117. n(n  1)(3n 2  19n  14) 12 124. 353 990 125. 317 90 n1 118. 126. n n 1 2 3