Sequence & Series PYQs Set 7 (PDF)

Set 7 of 10 · Questions 181-210 of 284

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sequence & Series · Set 7 (Q181-210)
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181. $Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and q respectively. Let d and D be the common differences of AP^{’} s in A and B respectively such that D = d + 3, d > 0 . If \frac{p + q}{p - q} = \frac{19}{5}, then p - q is equal to$ (1) $540$ (2) $450$ (3) $600$ (4) $630$ 182. $Let f : R \to R be such that for all x \in R (2^{1 + x} + 2^{1 - x}), f (x) and (3^{x} + 3^{- x}) are in A . P ., then the minimum value of f (x) is:$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 183. $If the sum of the second, third and fourth terms of a positive term G . P . is 3 and the sum of its sixth, seventh and.eighth terms is 243, then the sum of the first 50 terms of this G . P . is:$ (1) $\frac{2}{13} (3^{50} - 1)$ (2) $\frac{1}{13} (3^{50} - 1)$ (3) $\frac{1}{26} (3^{49} - 1)$ (4) $\frac{1}{26} (3^{50} - 1)$ 184. $The sum of the infinite series 1 + \frac{2}{3} + \frac{7}{3 ^{2}} + \frac{12}{3 ^{3}} + \frac{17}{3 ^{4}} + \frac{22}{3 ^{5}} + \dots is equal to$ (1) $\frac{9}{4}$ (2) $\frac{15}{4}$ (3) $\frac{13}{4}$ (4) $\frac{11}{4}$ 185. $Let 2^{nd}, 8^{th} and 44^{th}, terms of a non-constant A.P. be respectively the 1^{st}, 2^{nd} and 3^{rd} terms of G.P. If the first term of A.P. is 1 then the sum of first 20 terms is equal to-$ (1) $980$ (2) $960$ (3) $990$ (4) $970$ 186. $Let 3, a, b, c be in A.P. and 3, a - 1, b + 1, c + 9 be in G.P. Then, the arithmetic mean of a, b and c is :$ (1) $-4$ (2) $-1$ (3) $13$ (4) $11$ 187. $If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to$ (1) $-1080$ (2) $-1020$ (3) $-1200$ (4) $-120$ 188. $If the sum of the series \frac{1}{1 \cdot ( 1 + d )} + \frac{1}{( 1 + d ) ( 1 + 2 d )} + \dots + \frac{1}{( 1 + 9 d ) ( 1 + 10 d )} is equal to 5 , then 50 d is equal to :$ (1) $10$ (2) $5$ (3) $15$ (4) $20$ 189. $The third term of a G.P. is 9. The product of its first five terms is$ (1) $3^{5}$ (2) $3^{10}$ (3) $3^{9}$ (4) $3^{12}$ 190. $Suppose that the number of terms in an A.P. is 2 k, k \in N . If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27 , then k is equal to:$ (1) $6$ (2) $5$ (3) $8$ (4) $4$ 191. $Let the first three terms 2, p and q, with q \neq = 2, of a G.P. be respectively the 7^{th}, 8^{th} and 13^{th} terms of an A.P. If the 5^{th} term of the G.P. is the n^{th} term of the A.P., then n is equal to:$ (1) $163$ (2) $151$ (3) $177$ (4) $169$ 192. $If \frac{1}{1 ^{4}} + \frac{1}{2 ^{4}} + \frac{1}{3 ^{4}} + \dots \infty = \frac{π ^{4}}{90}, \frac{1}{1 ^{4}} + \frac{1}{3 ^{4}} + \frac{1}{5 ^{4}} + \dots \infty = α \frac{1}{2 ^{4}} + \frac{1}{4 ^{4}} + \frac{1}{6 ^{4}} + \dots \infty = β then \frac{α}{β} is equal to:$ (1) $23$ (2) $14$ (3) $18$ (4) $15$ 193. $The common difference of the A.P.: a_{1}, a_{2}, \dots, a_{m} is 13 more than the common difference of the A.P.: b_{1}, b_{2}, \dots, b_{n} . If b_{31} = - 277, b_{43} = - 385 and a_{78} = 327, then a_{1} is equal to$ (1) $16$ (2) $21$ (3) $19$ (4) $24$ 194. $In an arithmetic progression, if S_{40} = 1030 and S_{12} = 57, then S_{30} - S_{10} is equal to:$ (1) $525$ (2) $510$ (3) $515$ (4) $505$ 195. $The product of 2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \dots \infty =$ (1) $2$ (2) $2$ (3) $2^{\frac{1}{4}}$ (4) $1$ 196. $If each term of a geometric progression a_{1}, a_{2}, a_{3}, \dots with a_{1} = \frac{1}{8} and a_{2} \neq = a_{1}, is the arithmetic mean of the next two terms and S_{n} = a_{1} + a_{2} + \dots + a_{n}, then S_{20} - S_{18} is equal to$ (1) $2^{15}$ (2) $- 2^{18}$ (3) $2^{18}$ (4) $- 2^{15}$ 197. $Assertion (A) : (1) + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + \dots + (81 + 90 + 100) = 1000 Reason (R) : \sum_{r = 1}^{n} (r^{3} - [r - 1]^{3}) = n^{3} for any natural number n$ (1) $Both (A) and (R) are true and (R) is the correct explanation of (A)$ (2) $Both (A) and (R) are true but (R) is not correct explanation of (A)$ (3) $(A) is true but (R) is false$ (4) $(A) is false but (R) is true$ 198. $For all n \in N, if 1^{2} + 2^{2} + 3^{2} + \dots + n^{2} > x, then x =$ (1) $\frac{n ^{3}}{3}$ (2) $\frac{n ^{3}}{2}$ (3) $n^{3}$ (4) $\frac{n ^{4}}{4}$ 199. $\frac{6}{3 ^{26}} + \frac{10 \cdot 1}{3 ^{25}} + \frac{10 \cdot 2}{3 ^{24}} + \frac{10 \cdot 2 ^{2}}{3 ^{23}} + \dots + \frac{10 \cdot 2 ^{24}}{3} is equal to:$ (1) $2^{26}$ (2) $3^{25}$ (3) $3^{26}$ (4) $2^{25}$ 200. $If a_{1}, a_{2}, a_{3} ....and b_{1}, b_{2}, b_{3} .... are A.P. and a_{1} = 2, a_{10} = 3, a_{1} b_{1} = 1 = a_{10} b_{10} then a_{4} b_{4} is equal to$ (1) $\frac{35}{27}$ (2) $1$ (3) $\frac{27}{28}$ (4) $\frac{28}{27}$ 201. $A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after :$ (1) $18 months$ (2) $19 months$ (3) $20 months$ (4) $21 months$ 202. $Let x_{1}, x_{2},..., x_{100} be in an arithmetic progression, with x_{1} = 2 and their mean equal to 200. If y_{i} = i (x_{i} - i), 1 \leq i \leq 100, then the mean of y_{1}, y_{2}, ..., y_{10} is$ (1) $10100$ (2) $10101.50$ (3) $10049.50$ (4) $10051.50$ 203. $Let S_{n} denote the sum of first n-terms of an arithmetic progression. If S_{10} = 530, S_{5} = 140, then S_{20} - S_{6} is equal to :$ (1) $1862$ (2) $1842$ (3) $1852$ (4) $1872$ 204. $Let a, b, c and d be positive real numbers such that a + b + c + d= 11. If the maximum value of a^{5} b^{3} c^{2} d is 3750 β, then the value of β is$ (1) $90$ (2) $110$ (3) $55$ (4) $108$ 205. $If \sum r! (r^{3} + 6 r^{2} + 2 r + 5) = α (11!) then the value of α is equal to______$ 206. $If m arithmetic means (A . Ms) and three geometric means (G . Ms) are inserted between 3 and 243 such that 4^{th} A . M . is equal to 2^{nd} G . M ., then m is equal to$ 207. $Let α = 1^{2} + 4^{2} + 8^{2} + 13^{2} + 19^{2} + 26^{2} + \dots . upto 10 terms and β = \sum_{n = 1}^{10} n^{4} . If 4 α - β = 55 k + 40, then k is equal to$ 208. $Let A_{1}, A_{2}, A_{3}, \dots \dots be squares such that for each n \geq 1, the length of the side of A_{n} equals the length of diagonal of A_{n} + 1 . If the length of A_{1} is 12 cm, then the smallest value of n for which area of A_{n} is less than one, is$ 209. $The greatest integer less than or equal to the sum of first 100 terms of the sequence \frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, .... is equal to$ 210. $The sum of all the elements in the set {n ∈ {1, 2,…, , 100} H.C.F. of n and 2040 is 1} is equal to ____.$