JEE MATH APEX

Sequence and Series-JEE-Main-PYQ’s

Mathematics Questions

Let the arithmetic mean of \frac{1}{a} and \frac{1}{b} be \frac{5}{16}, a > 2. If \alpha is such that a, 4, \alpha, b are in A.P., then the equation \alpha x^2 - ax + 2(\alpha - 2b) = 0 has :
(A) one root in (1, 4) and another in (-2, 0)
(B) one root in (0, 2) and another in (-4, -2)
(C) both roots in the interval (-2, 0)
(D) complex roots of magnitude less than 2
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\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 :
(A) 2^{26}
(B) 3^{25}
(C) 3^{26}
(D) 2^{25}
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The value of \sum_{k=1}^{\infty} (-1)^{k+1} \left( \frac{k(k+1)}{k!} \right) is
(A) e/2
(B) \sqrt{e}
(C) 2/e
(D) 1/e
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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
(A) 21
(B) 19
(C) 24
(D) 16
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Let a_1, a_2, a_3, a_4 be an A.P. of four terms such that each term of the A.P. and its common difference l are integers. If a_1 + a_2 + a_3 + a_4 = 48 and a_1 a_2 a_3 a_4 + l^4 = 361, then the largest term of the A.P. is equal to
(A) 27
(B) 24
(C) 23
(D) 21
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(\frac{1}{3} + \frac{4}{7}) + (\frac{1}{3^2} + \frac{1}{3} \times \frac{4}{7} + \frac{4^2}{7^2}) + (\frac{1}{3^3} + \frac{1}{3^2} \times \frac{4}{7} + \frac{1}{3} \times \frac{4^2}{7^2} + \frac{4^3}{7^3}) + \dots upto infinite terms, is equal to
(A) \frac{7}{4}
(B) \frac{4}{3}
(C) \frac{6}{5}
(D) \frac{5}{2}
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Let 729, 81, 9, 1, \dots be a sequence and P_n denote the product of the first n terms of this sequence. If 2 \sum_{n=1}^{40} (P_n)^{\frac{1}{n}} = \frac{3^{\alpha}-1}{3^{\beta}} and \gcd(\alpha, \beta) = 1, then \alpha + \beta is equal to
(A) 73
(B) 74
(C) 75
(D) 76
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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
(A) 238
(B) 136
(C) 476
(D) 952
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Let \sum_{k=1}^n a_k = \alpha n^2 + \beta n. If a_{10} = 59 and a_6 = 7a_1, then \alpha + \beta is equal to :
(A) 3
(B) 5
(C) 7
(D) 12
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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
(A) -26
(B) -20
(C) -24
(D) -22
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