Sequence & Series PYQs Set 1 (PDF)

Set 1 of 10 · Questions 1-30 of 284

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sequence & Series · Set 1 (Q1-30)
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1. $A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of m is equal to:$ (1) $150$ (2) $180$ (3) $160$ (4) $125$ 2. $The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is \frac{27}{19} . Then the common ratio of this series is:$ (1) $\frac{1}{3}$ (2) $\frac{2}{3}$ (3) $\frac{2}{9}$ (4) $\frac{4}{9}$ 3. $If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms is$ (1) $-22$ (2) $-24$ (3) $-20$ (4) $-26$ 4. $If p (\frac{1}{q} + \frac{1}{r}), q (\frac{1}{r} + \frac{1}{p}), r (\frac{1}{p} + \frac{1}{q}) are in A.P., then p, q, r$ (1) $are in A.P.$ (2) $are not in A.P.$ (3) $are not in G.P.$ (4) $are in G.P.$ 5. $The sum 1 + \frac{1 ^{3} + 2 ^{3}}{1 + 2} + \frac{1 ^{3} + 2 ^{3} + 3 ^{3}}{1 + 2 + 3} + \dots \dots + \frac{1 ^{3} + 2 ^{3} + 3 ^{3} + \dots + 15 ^{3}}{1 + 2 + 3 + \dots + 15} - \frac{1}{2} (1 + 2 + 3 + \dots + 15 is equal to :$ (1) $620$ (2) $1240$ (3) $1860$ (4) $660$ 6. $In an increasing geometric progression of positive terms, the sum of the second and sixth terms is \frac{70}{3} and the product of the third and fifth terms is 49 . Then the sum of the 4^{th}, 6^{th} and 8^{th} terms is equal to :$ (1) $96$ (2) $91$ (3) $84$ (4) $78$ 7. $If α, β and γ are three consecutive terms of a non-constant G . P . such that the equations α x^{2} + 2 βx + γ = 0 and x^{2} + x - 1 = 0 have a common root, then α (β + γ) is equal to:$ (1) $0$ (2) $αβ$ (3) $αγ$ (4) $βγ$ 8. $If the first term of an A . P . is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A . P . is:$ (1) $\frac{1}{6}$ (2) $\frac{1}{4}$ (3) $\frac{1}{7}$ (4) $\frac{1}{5}$ 9. $Let S_{n} denote the sum of the first n terms of an arithmetic progression. If S_{10} = 390 and the ratio of the tenth and the fifth terms is 15 : 7, then S_{15} - S_{5} is equal to:$ (1) $800$ (2) $890$ (3) $790$ (4) $690$ 10. $Let S_{n} denote the sum of the first n terms of an A . P . If S_{4} = 16 and S_{6} = - 48 then S_{10} is equal to :$ (1) $- 260$ (2) $- 410$ (3) $- 320$ (4) $- 380$ 11. $Let a_{1}, a_{2}, a_{3} \dots be an A.P. If a_{7} = 3, the product a_{1} a_{4} is minimum and the sum of its first n terms is zero, then n! - 4 a_{n (n + 2)} is equal to?$ (1) $24$ (2) $\frac{33}{4}$ (3) $\frac{381}{4}$ (4) $9$ 12. $Let A = {n \in [100, 700] \cap N : n is neither a multiple of 3 nor a multiple of 4} . Then the number of elements in A is$ (1) $290$ (2) $280$ (3) $300$ (4) $310$ 13. $Let a_{1}, a_{2}, a_{3}, \dots . be a G.P. of increasing positive numbers. If a_{3} a_{5} = 729 and a_{2} + a_{4} = \frac{111}{4}, then 24 (a_{1} + a_{2} + a_{3}) is equal to$ (1) $128$ (2) $129$ (3) $131$ (4) $130$ 14. $Let S be the sum of the first 9 terms of the series: {x + ka} + {x^{2} + (k + 2) a} + {x^{3} + (k + 4) a} + {x^{4} + (k + 6) a} + \dots where a \neq = 0 and x \neq = 1 If S = \frac{x ^{10} - x + 45 a ( x - 1 )}{x - 1} then k is equal to :$ (1) $- 3$ (2) $1$ (3) $- 5$ (4) $3$ 15. $If S_{n} = \frac{1}{6 \cdot 11} + \frac{1}{11 \cdot 16} + \frac{1}{16 \cdot 21} + \dots to n terms, then 6 S_{n} equals$ (1) $\frac{5 n - 4}{5 n + 6}$ (2) $\frac{n}{5 n + 6}$ (3) $\frac{2 n - 1}{5 n + 6}$ (4) $\frac{1}{5 n + 6}$ 16. $If y = \frac{3}{4} + \frac{3 \cdot 5}{4 \cdot 8} + \frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12} + \dots \infty, then$ (1) $y^{2} - 2 y + 5 = 0$ (2) $y^{2} + 2 y - 7 = 0$ (3) $y^{2} - 3 y + 4 = 0$ (4) $y^{2} + 4 y - 6 = 0$ 17. $The value of lim_{n \to \infty} (\sum_{k = 1}^{n} \frac{k ^{3} + 6 k ^{2} + 11 k + 5}{( k + 3 ) !}) is:$ (1) $4/3$ (2) $2$ (3) $7/3$ (4) $5/3$ 18. $Let α and β be the roots of the quadratic equation x^{2} sin θ - x (sin θ cos θ + 1) + cos θ = 0 (0 < θ < 45^{\circ}) and α < β . Then \sum (α^{n} + \frac{( - 1 ) ^{n}}{β ^{n}}) is equal to :$ (1) $\frac{1}{1 - cos θ} - \frac{1}{1 + sin θ}$ (2) $\frac{1}{1 + cos θ} + \frac{1}{1 - sin θ}$ (3) $\frac{1}{1 - cos θ} + \frac{1}{1 + sin θ}$ (4) $\frac{1}{1 + cos θ} - \frac{1}{1 - sin θ}$ 19. $For positive integers n, if 4 a_{n} = (n^{2} + 5 n + 6) and S_{n} = \sum_{k = 1}^{n} (\frac{1}{a _{k}}), then the value of 507 S_{2025} is:$ (1) $540$ (2) $675$ (3) $1350$ (4) $135$ 20. $\sum_{r = 1}^{20} (r^{2} + 1) (r!) is equal to:$ (1) $22! - 21!$ (2) $22! - 2 (21!)$ (3) $21! - 2 (20!)$ (4) $21! - 20!$ 21. $Let a_{1}, a_{2}, \dots ., a_{30} be an A . P ., S = \sum a_{i} and T = \sum a_{(2 i - 1)} . If a_{5} = 27 and S - 2 T = 75 then a_{10} is equal to:$ (1) $52$ (2) $57$ (3) $47$ (4) $42$ 22. $If the 2^{nd} and 5^{th} terms of a G.P. are 24 and 3 respectively, then the sum of 1^{st} six terms is$ (1) $\frac{189}{5}$ (2) $\frac{2}{189}$ (3) $\frac{189}{2}$ (4) $\frac{179}{2}$ 23. $Let a_{1} = 1 and for n ⩾ 1, a_{n + 1} = \frac{1}{2} a_{n} + \frac{n ^{2} - 2 n - 1}{n ^{2} ( n + 1 ) ^{2}} . Then (\sum (a_{n} - \frac{2}{n ^{2}})) is equal to____.$ 24. $If 8 = 3 + \frac{1}{4} (3 + p) + \frac{1}{4 ^{2}} (3 + 2 p) + \frac{1}{4 ^{3}} (3 + 3 p) + \dots \infty, then the value of p is_____$ 25. $The number of 3 -digit numbers, that are divisible by 2 and 3 , but not divisible by 4 and 9 , is .$ 26. $Let S_{n} be the sum to n-terms of an arithmetic progression 3, 7, 11, \dots \dots If 40 < (\frac{6}{n ( n + 1 )} \sum_{k = 1}^{n} s_{k}) < 42, then n equals$ 27. $If the sum of the first 10 terms of the series \frac{4 \cdot 1}{1 + 4 \cdot 1 ^{4}} + \frac{4 \cdot 2}{1 + 4 \cdot 2 ^{4}} + \frac{4 \cdot 3}{1 + 4 \cdot 3 ^{4}} + \dots . is \frac{m}{n}, where gcd (m, n) = 1, then m + n is equal to$ 28. $The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7 is _______$ 29. $In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is R - (a, b), then a^{2} + b^{2} is equal to____.$ 30. $If \frac{1}{2 \times 3 \times 4} + \frac{1}{3 \times 4 \times 5} + \times \frac{1}{4 \times 5 \times 6} + ... + \frac{1}{100 \times 101 \times 102} = \frac{k}{101}, then 34k is equal to _______ .$