Sequence & Series PYQs Set 10 (PDF)

Set 10 of 10 · Questions 271-284 of 284

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sequence & Series · Set 10 (Q271-284)
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271. $Let a_{1}, a_{2}, a_{3}, \dots, a_{10} be in G.P. with a_{i} > 0 for i = 1, 2, \dots, 10 and S be the set of pairs (r, k), r, k \in N (the set of natural numbers) for which lo g_{e} (a_{1}^{r} a_{2}^{k}) lo g_{e} (a_{4}^{r} a_{5}^{k}) lo g_{e} (a_{7}^{r} a_{8}^{k}) lo g_{e} (a_{2}^{r} a_{3}^{k}) lo g_{e} (a_{5}^{r} a_{6}^{k}) lo g_{e} (a_{8}^{r} a_{9}^{k}) lo g_{e} (a_{3}^{r} a_{4}^{k}) lo g_{e} (a_{6}^{r} a_{7}^{k}) lo g_{e} (a_{9}^{r} a_{10}^{k}) = 0 . Then the number of elements in S is :$ (1) $4$ (2) $infinitely many$ (3) $2$ (4) $10$ 272. $Let Sn = \sum_{k = 1}^{4 n} (- 1)^{\frac{k ( k + 1 )}{2}} k2 . Then Sn can take value(s)$ (1) $1056$ (2) $1088$ (3) $1120$ (4) $1332$ 273. $The value of the sum 1 \times 2 \times 3 + 2 \times 3 \times 4 + 3 \times 4 \times 5 + \dots upto n terms is equal to$ (1) $\frac{1}{6} n^{2} (2 n^{2} + 1)$ (2) $\frac{1}{6} (n^{2} - 1) (2 n - 1) (2 n + 3)$ (3) $\frac{1}{8} (n^{2} + 1) (n^{2} + 5)$ (4) $\frac{1}{4} n (n + 1) (n + 2) (n + 3)$ 274. $The sum of first n terms of \frac{1}{2.5} + \frac{1}{5.8} + \frac{1}{8.11} + \dots is$ (1) $\frac{3 n}{2 ( 3 n + 2 )}$ (2) $\frac{3 n}{3 n + 2}$ (3) $\frac{n}{2 ( 3 n + 2 )}$ (4) $\frac{n}{3 n + 2}$ 275. $If x = \frac{1.3}{3.6} + \frac{1.3 . 5}{3.6 . 9} + \frac{1.3 . 5.7}{3.6 . 9.12} + \dots to infinite terms, then 9 x^{2} + 24 x =$ (1) $31$ (2) $11$ (3) $41$ (4) $21$ 276. $Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5. Let the sum of its first five terms be \frac{98}{25} . Then the sum of the first 21 terms of an AP, whose first term is 10ar, n^{t h} term is a_{n} and the common difference is 10 a r^{2}, is equal to :$ (1) $21 a_{11}$ (2) $22 a_{11}$ (3) $15 a_{16}$ (4) $14 a_{16}$ 277. $If [x] be the greatest integer less than or equal to x, then \sum_{n = 8}^{100} [\frac{( - 1 ) ^{n} n}{2}] is equal to:$ (1) $0$ (2) $4$ (3) $-2$ (4) $2$ 278. $The sum \sum_{n = 1}^{21} \frac{3}{( 4 n - 1 ) ( 4 n + 3 )} is equal to$ (1) $\frac{7}{87}$ (2) $\frac{7}{29}$ (3) $\frac{14}{87}$ (4) $\frac{21}{29}$ 279. $The 8^{t h} common term of the series S_{1} = 3 + 7 + 11 + 15 + 19 + \dots S_{2} = 1 + 6 + 11 + 16 + 21 + \dots is ___________$ 280. $is l, then l^{2} is equal to$ 281. $The series of positive multiples of 3 is divided into sets : {3}, {6, 9,12}, {15, 18, 21, 24, 27},... Then the sum of the elements in the set is equal to__________,$ 282. $If \sum_{k = 1}^{10} \frac{k}{k ^{4} + k ^{2} + 1} = \frac{m}{n}, where m and n are co-prime, then m + n is equal to$ 283. $\frac{2 ^{3} - 1 ^{3}}{1 \times 7} + \frac{4 ^{3} - 3 ^{3} + 2 ^{3} - 1 ^{3}}{2 \times 11} + \frac{6 ^{3} - 5 ^{3} + 4 ^{3} - 3 ^{3} + 2 ^{3} - 1 ^{3}}{3 \times 15} + ... + \frac{3 0 ^{3} - 2 9 ^{3} + 2 8 ^{3} - 2 7 ^{3} + ... + 2 ^{3} - 1 ^{3}}{15 \times 63} is equal to_____.$ 284. $If \frac{6}{3 ^{12}} + \frac{16}{3 ^{11}} + \frac{20}{3 ^{10}} + \frac{40}{3 ^{9}} + .... + \frac{10240}{3} = 2^{n} . m, where m is odd, then m.n is equal to_____$