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JEE Main Sequences and Series questions

58 Mathematics questions on Sequences and Series as asked in JEE Main, newest first. 33 come with a worked solution.

  1. Let f(x) be a function such that f(x + y) = f(x) · f(y) for all x, y ∈ N. If f(1) = 3 and sum_k=1^n f(k) = 3279, then the value of n is

    2023

  2. For three positive integers p, q, r, x^pq^2 = y^qr = z^p^2r and r = pq + 1 such that 3, 3log_y x, 3log_z y, 7log_x z are in A.P. with common difference 1/2. Then r - p - q is equal to

    2023

  3. If dfrac1^3 + 2^3 + 3^3 + … up to n terms1·3 + 2·5 + 3·7 + … up to n terms = 9/5, then the value of n is

    2023 · with solution

  4. Suppose f is a function satisfying f(x+y) = f(x) + f(y) for all x, y ∈ N and f(1) = 1/5. If sum_n=1^m (f(n))/(n(n+1)(n+2)) = 1/12, then m is equal to _______

    2023 · with solution

  5. For the two positive numbers a, b, if a, b and 1/18 are in a geometric progression, while 1/a, 10 and 1/b are in an arithmetic progression, then 16a + 12b is equal to ___________.

    2023 · with solution

  6. Let a_1, a_2, a_3, … be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then a_1 a_9 + a_2 a_4 a_9 + a_5 + a_7 is equal to ________.

    2023 · with solution

  7. The 4^th term of GP is 500 and its common ratio is 1/m, m ∈ N. Let S_n denote the sum of the first n terms of this GP. If S_6 > S_5 + 1 and S_7 < S_6 + 1/2, then the number of possible values of m is

    2023 · with solution

  8. Let A_1, A_2, A_3 be the three A.P. with the same common difference d and having their first terms as A, A+1, A+2, respectively. Let a, b, c be the 7^th, 9^th, 17^th terms of A_1, A_2, A_3, respectively such that…

    2023

  9. Let a, b and c be in GP with common ratio r, where a ≠ 0 and 0 < r ≤ 1/2. If 3a, 7b and 15c are the first three terms of an AP, then the 4th term of this AP is

    2019 · with solution

  10. If a_1, a_2, a_3, …, a_n are in AP and a_1 + a_4 + a_7 + … + a_16 = 114, then a_1 + a_6 + a_11 + a_16 is equal to

    2019 · with solution

  11. The sum of series 1 + (1^3 + 2^3)/(1 + 2) + (1^3 + 2^3 + 3^3)/(1 + 2 + 3) + … + (1^3 + 2^3 + 3^3 + … + 15^3)/(1 + 2 + 3 + … + 15) - 1/2(1 + 2 + 3 + … + 15) is equal to

    2019 · with solution

  12. The sum sum_k=1^20 k · 1/(2^k) is equal to

    2019

  13. If 19th term of a non-zero AP is zero, then its (49th term) : (29th term) is

    2019 · with solution

  14. Let S_n denote the sum of the first n terms of an AP. If S_4 = 16 and S_6 = -48, then S_10 is equal to

    2019 · with solution

  15. The sum of series (3 × 1^3)/(1^2) + (5 × (1^3 + 2^3))/(1^2 + 2^2) + (7 × (1^3 + 2^3 + 3^3))/(1^2 + 2^2 + 3^2) + … upto 10th term, is

    2019

  16. Let S_n = 1 + q + q^2 + ... + q^n and T_n = 1 + ((q+1)/2 ) + ((q+1)/2 )^2 + ... + ((q+1)/2 )^n, where q is a real number and q ≠ 1. If ^101C_1 + ^101C_2 · S_1 + ... + ^101C_101 · S_100 = α T_100, then α is equal to

    2019

  17. The product of three consecutive terms of a GP is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an AP. Then, the sum of the original three terms of the given GP is

    2019 · with solution

  18. The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is 27/19. Then, the common ratio of this series is

    2019

  19. If the sum and product of the first three terms in an AP are 33 and 1155, respectively, then a value of its 11th term is

    2019 · with solution

  20. Let a_1, a_2, …, a_10 be a GP. If a_3/a_1 = 25, then a_9/a_5 equals

    2019 · with solution

  21. Let S_k = (1+2+3+…+k)/k. If S_1^2 + S_2^2 + … + S_10^2 = 5/12A, then A is equal to

    2019

  22. Let the sum of the first n terms of a non-constant AP a_1, a_2, a_3, … be 50n + (n(n-7))/2A, where A is a constant. If d is the common difference of this AP, then the ordered pair (d, a_50) is equal to

    2019 · with solution

  23. If a, b and c be three distinct real numbers in GP and a + b + c = xb, then x cannot be

    2019 · with solution

  24. The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is

    2019 · with solution

  25. Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression fracx^m y^n(1 + x^2m)(1 + y^2n) is

    2019

  26. The sum of the following series 1 + 6 + (9(1^2 + 2^2 + 3^2))/7 + (12(1^2 + 2^2 + 3^2 + 4^2))/9 + (15(1^2 + 2^2 + ·s + 5^2))/11 + … up to 15 terms is

    2019

  27. Let a_1, a_2, …, a_30 be an AP, S = sum_i=1^30 a_i and T = sum_i=1^15 a_(2i-1). If a_5 = 27 and S - 2T = 75, then a_10 is equal to

    2019 · with solution

  28. If 5, 5r, 5r² are the lengths of the sides of a triangle, then r cannot be equal to

    2019 · with solution

  29. Let a_1, a_2, a_3, …, a_49 be in AP such that sum_k=0^12 a_4k+1 = 416 and a_9 + a_43 = 66. If a_1^2 + a_2^2 + … + a_17^2 = 140 m, then m is equal to

    2018

  30. Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series 1^2 + 2 · 2^2 + 3^2 + 2 · 4^2 + 5^2 + 2 · 6^2 + … If B - 2A = 100λ, then λ is equal to

    2018

  31. For any three positive real numbers a, b and c, if 9(25a^2 + b^2) + 25(c^2 - 3ac) = 15b(3a + c), then

    2017

  32. If the 2nd, 5th and 9th terms of a non-constant AP are in GP, then the common ratio of this GP is

    2016

  33. Let b_i > 1 for i = 1, 2, …, 101. Suppose log_e b_1, log_e b_2, …, log_e b_101 are in AP with the common difference log_e 2. Suppose a_1, a_2, …, a_101 are in AP such that a_1 = b_1 and a_51 = b_51. If t = b_1 + b_2 + ……

    2016

  34. If the sum of the first ten terms of the series (13/5 )^2 + (22/5 )^2 + (31/5 )^2 + 4^2 + (44/5 )^2 + … is 16/5m, then m is equal to

    2016

  35. If m is the AM of two distinct real numbers l and n (l, n > 1) and G_1, G_2 and G_3 are three geometric means between l and n, then G_1^4 + 2G_2^4 + G_3^4 equals

    2015

  36. The sum of first 9 terms of the series (1^3)/1 + (1^3 + 2^3)/(1+3) + (1^3 + 2^3 + 3^3)/(1+3+5) + … is

    2015 · with solution

  37. Three positive numbers form an increasing GP. If the middle term in this GP is doubled, then new numbers are in AP. Then, the common ratio of the GP is

    2014 · with solution

  38. If (10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + … + 10(11)^9 = k(10)^9, then k is equal to

    2014

  39. The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, ... is

    2013

  40. If a_1, a_2, a_3, … are in a harmonic progression with a_1 = 5 and a_20 = 25. Then, the least positive integer n for which a_n < 0, is

    2012 · with solution

  41. If the sum of first n terms of an AP is cn^2, then the sum of squares of these n terms is

    2009 · with solution

  42. Let f(x) = ax^2 + bx + c, a ≠ 0 and Δ = b^2 - 4ac. If α + β, α^2 + β^2 and α^3 + β^3 are in GP, then

    2005

  43. An infinite GP has first term x and sum 5, then x belongs to

    2004 · with solution

  44. Let a, b, c be in an AP and a^2, b^2, c^2 be in GP. If a < b < c and a + b + c = 3/2, then the value of a is

    2002 · with solution

  45. If a_1, a_2, …, a_n are positive real numbers whose product is a fixed number c, then the minimum value of a_1 + a_2 + … + a_n-1 + 2a_n is

    2002 · with solution

  46. If the positive numbers a, b, c, d are in AP. Then, abc, abd, acd, bcd are

    2001

  47. Let α, β be the roots of x^2 - x + p = 0 and γ, δ be the roots of x^2 + 4x + q = 0. If α, β, γ, δ are in GP, then the integer values of p and q respectively are

    2001

  48. If the sum of the first 2n terms of the AP series 2, 5, 8, … is equal to the sum of the first n terms of the AP series 57, 59, 61, …, then n equals

    2001 · with solution

  49. Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and the second term is 3/4, then

    2000 · with solution

  50. Let a_1, a_2, …, a_10 be in AP and h_1, h_2, equal to …, h_10 be in HP. If a_1 = h_1 = 2 and a_10 = h_10 = 3, then a_4 h_7 is

    1999 · with solution

  51. The harmonic mean of the roots of the equation (5 + √(2))x^2 - (4 + √(5))x + 8 + 2√(5) = 0 is

    1999 · with solution

  52. If x > 1, y > 1, z > 1 are in GP, then 1/(1 + ln x), 1/(1 + ln y), 1/(1 + ln z) are in

    1998 · with solution

  53. The product of n positive numbers is unity, then their sum is

    1991

  54. If a, b and c are distinct positive numbers, then the expression (b+c-a)(c+a-b)(a+b-c)-abc is

    1991 · with solution

  55. If a, b, c, d and p are distinct real numbers such that (a^2 + b^2 + c^2) p^2 - 2(ab + bc + cd) p + (b^2 + c^2 + d^2) ≤ 0, then a, b, c, d

    1987 · with solution

  56. If a, b, c are in GP, then the equations ax^2 + 2bx + c = 0 and dx^2 + 2ex + f = 0 have a common root, if d/a, e/b, f/c are in

    1985

  57. The third term of a geometric progression is 4. The product of the first five terms is

    1982 · with solution

  58. If x_1, x_2, …, x_n are any real numbers and n is any positive integer, then

    1982

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