Sequence & Series PYQs Set 9 (PDF)

Set 9 of 10 · Questions 241-270 of 284

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sequence & Series · Set 9 (Q241-270)
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241. $Let a_{n} be n^{t h} term of the series 5 + 8 + 14 + 23 + 35 + 50+.......and S_{n} = \sum_{k = 1}^{n} a_{k} . Then S_{30} - S_{40} is equal to$ (1) $11310$ (2) $11260$ (3) $11290$ (4) $11280$ 242. $Let A_{1}, A_{2}, A_{3}, ...... be an increasing geometric progression of positive real numbers. If A_{1} A_{3} A_{5} A_{7} = \frac{1}{1296} and A_{2} + A_{4} = \frac{7}{36}, then, the A_{6} + A_{8} + A_{10} value of A_{6} + A_{8} + A_{10} is equal to$ (1) $33$ (2) $37$ (3) $43$ (4) $47$ 243. $The sum of first 20 terms of the sequence 0.7, 0.77, 0.777,....., is$ (1) $\frac{7}{81} (179 - 10-20)$ (2) $\frac{7}{9} (99 - 10-20)$ (3) $\frac{7}{81} (179 + 10-20)$ (4) $\frac{7}{9} (99 + 10-20)$ 244. $If the sum of the first ten terms of the series (1 \frac{3}{5})^{2} + (2 \frac{2}{5})^{2} + (3 \frac{1}{5})^{2} + 4^{2} + (4 \frac{4}{5})^{2} + ...... is \frac{16}{5} m, then m is equal to :$ (1) $101$ (2) $100$ (3) $99$ (4) $102$ 245. $If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is$ (1) $21$ (2) $22$ (3) $23$ (4) $24$ 246. $n^{th} term of the series 1 + \frac{3}{7} + \frac{5}{7 ^{2}} + \frac{7}{7 ^{3}} + \dots is$ (1) $\frac{2 n - 1}{7 ^{n}}$ (2) $\frac{2 n - 1}{7 ^{n - 1}}$ (3) $\frac{2 n + 1}{7 ^{n - 1}}$ (4) $\frac{2 n + 1}{7 ^{n}}$ 247. $Consider an A.P.: a_{1}, a_{2}, \dots, a_{n}; a_{1} > 0 . If a_{2} - a_{1} = \frac{- 3}{4}, a_{n} = \frac{1}{4} a_{1}, and \sum_{i = 1}^{n} a_{i} = \frac{525}{2}, then \sum_{i = 1}^{17} a_{i} is equal to$ (1) $136$ (2) $476$ (3) $238$ (4) $952$ 248. $The value of \sum_{k = 1}^{\infty} (- 1)^{k + 1} (\frac{k ( k + 1 )}{k !}) is$ (1) $2/ e$ (2) $1/ e$ (3) $e /2$ (4) $e$ 249. $The sum \sum_{k = 1}^{20} k \frac{1}{2 ^{k}} is equal to:$ (1) $2 - \frac{3}{2 ^{17}}$ (2) $1 - \frac{11}{2 ^{20}}$ (3) $2 - \frac{11}{2 ^{19}}$ (4) $2 - \frac{21}{2 ^{20}}$ 250. $If n = 2020!, then \frac{1}{l o g _{2} n} + \frac{1}{l o g _{3} n} + \frac{1}{l o g _{4} n} + \dots + \frac{1}{l o g _{2020} n} =$ (1) $2020$ (2) $0$ (3) $2020$ (4) $1$ 251. $Let a_{1}, a_{2}, a_{3}, \dots be a G.P. such that a_{1} < 0, a_{1} + a_{2} = 4 and a_{3} + a_{4} = 16 . If \sum_{i = 1}^{9} a_{i} = 4 λ, then λ is equal to:$ (1) $- 513$ (2) $- \frac{511}{3}$ (3) $- 171$ (4) $171$ 252. $The parabolas a x^{2} + 2 b x + cy = 0 and d x^{2} + 2 e x + f y = 0 intersect on the line y = 1 . If a, b, c, d, e, f are positive real numbers and a, b, c are in G.P., then$ (1) $d, e, f are in A.P$ (2) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c} are in G.P$ (3) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c} are in A.P$ (4) $d, e, f are in G.P$ 253. $Let a_{1}, a_{2}, a_{3}, \dots be in an A.P. such that \sum_{k = 1}^{12} a_{2 k - 1} = - \frac{72}{5} a_{1}, a_{1} \neq = 0 . If \sum_{k = 1}^{n} a_{k} = 0, then n is:$ (1) $18$ (2) $17$ (3) $11$ (4) $10$ 254. $Let a_{n} be the n^{th} term of a G.P. of positive terms. If \sum_{n = 1}^{100} a_{2 n + 1} = 200 and \sum_{n = 1}^{100} a_{2 n} = 100, then \sum_{n = 1}^{200} a_{n}$ (1) $175$ (2) $150$ (3) $300$ (4) $225$ 255. $Let a, b, c > 1, a^{2}, b^{2} and c^{2} be in A.P., and lo g_{3} b, lo g_{e} a and lo g_{e} c be in G.P. If the sum of first 20 terms of an A.P., whose first term is \frac{a + 4 b + c}{3} and the common difference is \frac{a - 8 b + c}{10} is - 444 . Then ab c is equal to$ (1) $343$ (2) $216$ (3) $\frac{343}{8}$ (4) $\frac{125}{8}$ 256. $For three positive integers p, q, r, x^{p q^{2}} = y^{q r} = z^{p^{2} r} and r = pq + 1 such that 3, 3 lo g_{y} x, 3 lo g_{z} y, 7 lo g_{x} z are in A.P. with common difference \frac{1}{2}, then r - p - q is equal to$ (1) $2$ (2) $6$ (3) $12$ (4) $- 6$ 257. $If 0 < θ, φ < \frac{π}{2}, x = \sum_{n = 0}^{\infty} cos^{2 n} θ, y = \sum_{n = 0}^{\infty} sin^{2 n} φ and z = \sum_{n = 0}^{\infty} cos^{2 n} θ \cdot sin^{2 n} φ then$ (1) $xyz = 4$ (2) $xy - z = (x + y) z$ (3) $xy + yz + zx = z$ (4) $xy + z = (x + y) z$ 258. $Let y = f (x) represent a parabola with focus (- \frac{1}{2}, 0) and directrix y = - \frac{1}{2} . Then S = {x \in R : tan^{- 1} (f (x)) + sin^{- 1} (f (x) + 1) = \frac{π}{2}}$ (1) $contains exactly two elements$ (2) $Contains exactly one element$ (3) $Is an infinite set$ (4) $Is an empty set$ 259. $3 + 5 + 7 + \dots to n terms is$ (1) $n^{2}$ (2) $n (n + 2)$ (3) $n (n - 2)$ (4) $(n + 1)^{2}$ 260. $If lo g_{3} 2, lo g_{3} (2^{x} - 5), lo g_{3} (2^{x} - \frac{7}{2}) are in an arithmetic progression, then the value of x is equal to$ 261. $If (20)^{19} + 2(21)(20)^{18} + 3(21) (20)^{17} +... +20(21)^{19} = k(20)^{19}, then k is equal to_____.$ 262. $The sum to 20 terms of the series 2. 2^{2} - 3^{2} + 2. 4^{2} - 5^{2} + 2. 6^{2} -............is equal to___________.$ 263. $Let a1, a2, a3, ....., a11 be real numbers satisfying a1 = 15, 27 - 2a2 > 0 and ak = 2ak-1 - ak-2 for k = 3, 4, ...., 11. If \frac{a _{1}^{2} + a _{2}^{2} + .... + a _{1}^{2} _{1}}{11} = 90, then the value of \frac{a _{1} + a _{2} + ... + a _{11}}{11} is equal to$ 264. $If a_{1} (> 0), a_{2}, a_{3}, a_{4}, a_{5} are in a G.P., a_{2} + a_{4} = 2 a_{3} + 1 and 3 a_{2} + a_{3} = 2 a_{4}, then a_{2} + a_{4} + 2 a_{5} is equal to$ 265. $Let Sk, k = 1, 2,...., 100, denote the sum of the infinite geometric series whose first term is \frac{k -1}{k !} and the common ratio is \frac{1}{k} . Then the value of \frac{10 0 ^{2}}{100 !} + \sum_{k = 1}^{100} (k^{2} - 3 k + 1) S_{k} is$ 266. $Let (a_{k}) and (b_{k}), k \in N, be two G.P.s with common ratio r_{1} and r_{2} respectively such that a_{1} = b_{1} = 4 and r_{1} < r_{2} . Let c_{k} = a_{k} + b_{k}, k \in N . If c_{2} = 5 and c_{3} = \frac{13}{4} then \sum_{k = 1}^{\infty} c_{k} - (12 a_{6} + 8 b_{4}) is equal to _____$ 267. $Suppose a_{1}, a_{2}, 2, a_{3}, a_{4} be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is \frac{49}{2}, then a_{4} is equal to______________$ 268. $Let \sum_{n = 0}^{\infty} \frac{n ^{3} ( ( 2 n )! ) + ( 2 n - 1 ) ( n ! )}{( n ! ) ( ( 2 n )! )} = a e + \frac{b}{e} + c Where a, b, c \in Z and e = \sum_{n = 0}^{\infty} \frac{1}{n !} Then a^{2} - b + c is equal to __________$ 269. $Let A_{1}, A_{2}, A_{3} be the three AP. with the same common difference d having their first terms as A, A + 1, A + 2, respectively. Let a, b, c be 7^{t h}, 9^{t h}, 1 7^{t h} terms of A_{1}, A_{2}, A_{3}, respectively such that a 2 b c 71717111 + 70 = 0$ 270. $If \sum_{r = 1}^{25} (\frac{r}{r ^{4} + r ^{2} + 1}) = \frac{p}{q}, where p and q are positive integers such that g cd (p, q) = 1, then p + q is equal to ____.$