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Class 10 CBSE Airthmetic progressions important questions 2018 : Mathematics

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Abhilash Reddy
Gitanjali School, Hyderabad

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ARITHMETIC PROGRESSION (Key Points) Arithmetic progression (A.P.) :- An A.P. is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term. This fixed number is called the common difference of the A.P. If a is first term and d is common difference of an A.P. , then the A.P is a , a+d , a+2d , 2+3d .. The term of an a.p is denoted by and = a+(n-1) d , where a = first term and d = common difference. term from the end = l (n-1) d , where l = last term. Three terms a-d , a , a+d are in A.P with common difference d. Four terms a-3d , a-d , a+d ,a+3d are in A.P with common diff. 2d . The sum of first n natural number is The sum of n terms of an A.P with first term a and common difference d is denoted by ( +1) 2 = 2 { 2a+(n-1) d } also , = (a+l) where , l = last term. 2 = - 1 . Where = term of an A.P D = - 1 . Where d = common difference of an A.P. Problems for self evaluation. 1. Show that the sequence defined by =4 +7 is an AP. 2. Find the number of terms for given AP :7,13 ,19,25, ..,205. 3. The 7th term of an AP is 32 and it 13th term is 62. Find AP. 4. Find the sum of all two digit odd positive nos. 5. Find the value of x for AP. 1+6+11+16+ .+X=148. 6. Find the 10th term from the end of the AP 8,10,12, 126. 7. The sum of three numbers of AP is 3 and their product is -35.Find the numbers. 8. A man repays a loan of Rs3250 by paying Rs20 in the first month and then increase the payment by Rs15 every month .How long will it take him to clear the loan ? 9. The ratio of the sums of m and n terms of an AP is 2 : 2 .show that the ratio of the mth and nth terms is (2m-1) : (2n-1). 10. In an AP , the sum of first n terms is 3 2 2 + 5 2 , Find it 25th term. Arithmetic progressions Assignment 1. Which term of the A.P. 4,9,14,19...........is 124? 2. Find the middle term of the A.P. 6,13,20......216? 3. In the A.P. 1,7,13,19.......415. Prove that the sum of the middle terms is equal to the first and last terms. 4. If the 8th term of an A.P is 31 and the 15th term is 16 more than the 11th term of the A.P. Find A.P 5. The 10th term of A.P is 52 and 16th term is 82.Find 32nd term and general term. 6. If five times of 5th term of A.P. is equal to 8 times its 8th term. Show that its 13th term is zero. 7. How many two digit numbers are divisible by 7? 8. How many three digit numbers are divisible by 7? 9. How many multiples of 4 lie between 10 and 250? 10. How many two digit numbers are divisible by 3? 11. Find the number of integers between 20 and 500 which are divisible by 7? 12. If the term of an A.P is 1 and term be 1 .Show that its term is 1. 13. If the term of an A.P is q and term is p .Prove that its term is ( + ) 14. If m times the term of an A.P is equal to n times its term. Show that the ( + ) term is zero. 15. Which term of the sequence 20, 1914 , 1812 , 1734 ,.......is first negative term? 16. Find a,b and c if it is given that the numbers a,7,b,23,c are in A.P? 17. The 8th term of an A.P. is half the 2nd term and 11th term exceeds one third of its 4th term by 1.Find the 15th term? 18. Which term of A.P. 8,14,20,26....will be 72 more than its 41st term? 19. Find the term of A.P. 9,12,15,18.........which is 39 more than its 36th term? 20. Find 8th term from end of A.P.7, 10, 13 .........184? 21. The sum of 4th and 8th term of A.P. is 24 and sum of 6th and 10th term is 44.Find the A.P? 22. The 9th term of an A.P is equal to 6times its 2nd term. If its 5th term is 22.Find the A.P? 23. Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5. 24. If ( + 1) term of an A.P is twice the ( + 1) term. Prove that (3 + 1) term is twice the ( + + 1) term. 25. If 2 , + 10,3 + 2 are in A.P. find the value of ? 26. The sum of three numbers in A.P is -3 and their product is 8.Find the numbers. 27. Find four numbers in A.P whose sum is 20 and the sum of whose squares is 120. 28. If 1 = 3; = 3 1 + 1 for all n>1 .Find the first four terms of the sequence. 29. If numbers a,b,c,d,e form an A.P.then find the value of 4 + 6 4 + ? 30. The sum of first three terms of an A.P is 33.If the product of 1st and 3rd term exceeds the 2nd term by 29.Find the A.P 31. Determine so that 2 + 4 + 8, 2 2 + 3 + 6, 3 2 + 4 + 4 are three consecutive term of A.P. 32. The angles of a triangle are in A.P .The greatest angle is twice the least .Find all the angles. 33. Split 207 into three parts such that these are in A.P. and the product of two smaller parts is 4623. 34. If the term of an A.P is (2 + 1) ,find the sum of first terms of the A.P? 35. The sum of first six terms of an A.P is 42.The ratio of its 10th term to its 30th term is 1:3.Calculate the 1st and 13th term of the A.P. 36. 37. 38. 39. Find the sum of all 3 digit natural numbers which are divisible by 7? How many terms of A.P 54,51,48.....be taken so that their sum is 513? If = 5 2 + 3 ,then find its term? An A.P consists of 37 terms .The sum of three middle most terms is 225 and the sum of last three terms is 429.Find A.P? 40. If the term of an A.P is 1 and term be 1 1 .Show that sum of terms is ( + 1) 2 41. If the sum of terms of an A.P is same as the sum of terms .Show that the sum of its ( + ) term is zero? 42. The ratio of the sum of and terms of an A.P is 2 : 2 .Show that ratio of term is (2 1): (2 1). 43. Solve 1 + 4 + 7 + 10 + . . + = 287 44. A man repays a loan of 3250 by paying 20 in the first month and then increases the payment by 15 every month. How long will it take him to clear the loan? 45. Find the sum of the following A.Ps : 9 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. i)3, , 6, 2 15 2 2 . 25 ( 2 2 ), ( 2 ii) + , , 3 2 iii) ( ) , + + ) . Find the sum of last 10 terms of the A.P. 8,10,12,14........126. How many terms of A.P 18,16,14.......be taken so that their sum is zero? How many terms of A.P27,24,21............be taken so that their sum is zero? Find the sum of first 15 multiples of 8? Find the sum of first 40 positive integers divisible by a)3 b)5 c) 6. Find the sum of all 3-digit natural numbers which are multiples of 11. The first terms of an A.P is 2 and the last term is 50.The sum of all these terms is 442.Find the common difference? If the sum of 7 terms of an A.P is 49 and that of 17 terms is 289.Find the sum of n terms? `In an A.P the 1st term is 22 term is -11 and the sum of first n terms is 66 .Find n and d. The 1st and last term of an A.P are 5 and 45 .If sum of all its terms is 400.Find its common difference. The sum of q terms of an A.P is 162.The ratio of its 6th term to its 13th term is 1:2.Find first and 15th term of A.P? If nth term of an A.P. is given by an = 5n 3. Find the sum of its 50 terms. Sum of n terms of a sequence is given by = 2 2 2 If it is an A.P. then find its 20th term. Find the sum of the following series: 10 + 14 + 18 + 22 + 104 If the sum of first 13 terms of an A.P. is 91 and sum of its first 30 terms is 465. Find the sum of its 50terms. The sum of three numbers which are in A.P. is 9 and sum of their squares is 35. Find the numbers. Divide 32 into 4 parts such that they are in AP and the ratio of the product of first term and fourth terms = product of second and third terms is equal to 7/15. If the , , terms of an AP are a,b,c then prove a(q r) + b(r-p) = c (q p) The sum of n terms of an A.P. is given by = 5 2 3 , find the term of A.P. How many terms of the A.P. -6, -11/2, -5, are needed to give the sum -25? Find the 30th and 60th terms of the following sequence: (1) 13, 18, 23, 28, 33, .(2) -10, -7, 4, -1 . Find the sum of the following series:1. 3+8+13+18+23 .248. 2. -10, -15, -20, -25, .105 67. The sum of three numbers which are in A.P. is 9 and sum of their squares is 35. Find the numbers. 68. term of a sequence is given by the formula = 10 3 ,find the sum of its 20 terms. 69. The 8th term of an A.P. is 32 and its 12th term is 52. Find the A.P. 70. Write next two terms of given A.P. (a) -3, -5, -7,-9 .. 71. Find 10th term of given A.P. 10, 20, 30, 40 72. Find 20th term of A.P. whose first term = 10, c.d = 3 73. Find n, if the given value of x is nth term of A.P.17, 22, 27, 32 x = 267 74. 3 times the tenth term is equal to 5 times the twentieth term. Find twentieth term 75. Find the sum 2+4+6+ +202 76. Write sequence with nth term 5+2 n. Find sum of first 20 terms. 77. Third term of an A.P. is 21 and the eighth term is 56.Find A.P. and also find its eleventh term. 78. Find sum of all natural numbers between 1 and 98 which are multiples of 6. 79. How many terms of the series must be added to get sum 55 ,15, 13, 11, 9, 80. Sum of terms of A.P. is 36 and product is 1296. Find A.P 81. Determine k so that k + 2, 4k - 6 and 3k - 2 are three consecutive terms of an AP. 82. The first, second and the last terms of an AP are p, q and 2p respectively. Show that its sum is [3pq]/[2(q-p)]. 83. A circle is completely divided into n sectors in such a way that the angles of the sectors are in arithmetic progression. If the smallest-of these angles is 8 and the largest 72 , calculate n and the angle in the fourth sector. 84. Which term of AP: 3, 10, 17 ... will be 84 more than its 13th term? 85. If 9th term of an AP is zero, prove that 29th term is double the 19th term. 86. Find a, b such that 27, a, b - 6 are in A.P. 87. For what value of n, the terms of the sequences 3, 10, 17,... and 63, 65, 67,... are equal. 88. Find the sum of all odd integers between 78 and 500 which are divisible by 7. 89. Find n, if the given value of is term of A.P. 17, 22, 27, 32, ........ x = 267 90. Find the sum of all the odd numbers between 100 and 200 91. If 10 times the 10th term of an AP is equal to 15 times its 15th term, show that its 25th term is zero. 92. Find the sum 2+4+6+. . . +202 93. How many terms are there in the A.P -1,-5/6,-2/3,-1/2 .10/3? Also find its general term? 94. The 5th term of an AP is 24 and its 15th term is 74. Find the sum of its first 10 terms. 95. If the first term and last term of an AP are a and l respectively and its sum is S , prove that the common difference of the AP is equal to (l2 a2) / [2S-(l+a)] . 96. If the difference between the 21st and 10th terms of an AP is 55, find the difference between the 45th and 40th terms. 97. Find three numbers in an A.P. whose sum is 15 and product 80 98. How many terms of A.P. 22, 20, 18, . . . . . . . . should be taken so that their sum is zero? 99. Find the sum of odd positive integers less than 199. 100. How many two digits numbers between 3 and 102 are divisible by 6? 101. If 7 times the 7th term is equal to 11 times the 11th term of an A.P. Find its 18th term. 102. Which term of A.P. 13, 21, 29, . . . . . . will be 48 less than its 19th term? 103. Find the A.P. whose 3rd term is 13 and 6th term is +2. 104. Find the A.P., whose 5th term is 23 and 9th term is 43. 105. The angles of a triangle are in A.P. If the smallest angle is one fifth the sum of other two angles. Find the angles. 106. `Aditi saved Rs. 500 in the first month of a year and then increased her monthly savings by Rs. 50. If in the nth month, her monthly savings become Rs 1000. Find the value of 'n'. 107. The sum of first n terms of an A.P. is 2n2 + n . Find nth term and common deference of the A.P. 108. The sum of 3rd and 7th terms of an A.P. is 14 and the sum of 5th and 9th terms is 34. Find the first term and common difference of the A.P. 109. Find the sum of the first 30 terms of an A.P., whose nth term is 2 3n. If mth and nth terms of an A.P. are 1/n and1/m respectively, then find the sum of mn terms 110. If mth , nth and rth terms of an A.P. are x, y and z respectively, then prove that :- m( y z) + n(z x) + r (x y) = 0 111. If the roots of the equation a(b c) x2 + b(c a) x + c (a b) = 0 are equal, then show that 1/a , 1/b , 1/c are in A.P.

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