Class 11 Mathematics Sequences and Series MCQs Set 10

Question. If the first term of an A.P is –1 and common difference is – 3, then 12th term is
(a) 34
(b) 32
(c) –32
(d) –34
Answer: (d) –34

Question. If the sum to n terms of an A.P. is \( 3n^2 + 5n \) while \( T_m = 164 \), then value of m is
(a) 25
(b) 26
(c) 27
(d) 28
Answer: (c) 27

Question. Let \( T_r \) be the rth term of an AP for r=1, 2, ... If for some positive integers m and n we have \( T_m = 1/n \) and \( T_n = 1/m \), the \( T_{mn} = \)
(a) \( -1/mn \)
(b) \( 1/m + 1/n \)
(c) 1
(d) 0
Answer: (c) 1

Question. The interior angles of a polygon are in A.P. If the smallest angle is \( 100^\circ \) and the common difference is \( 4^\circ \), then the number of sides is
(a) 5
(b) 7
(c) 36
(d) 44
Answer: (a) 5

Question. If a, b, c, d, e, f are in A.P., then e–c is equal to
(a) 2(c – a)
(b) 2(d – c)
(c) f – e
(d) d – c
Answer: (b) 2(d – c)

Question. If the ratio between the sums of n terms of two A.P.’s is \( 3n + 8 : 7n + 15 \), then the ratio between their 12th terms is
(a) 16 : 7
(b) 7 : 16
(c) 74 : 169
(d) 169 : 74
Answer: (b) 7 : 16

Question. If the sum of the first ten terms of an A.P is four times the sum of its first five terms, then ratio of the first term to the common difference is
(a) 1 : 2
(b) 2 : 1
(c) 1 : 4
(d) 4 : 1
Answer: (a) 1 : 2

Question. If \( S_n \) denotes the sum of n terms of an A.P., then \( S_{n+3} - 3S_{n+2} + 3S_{n+1} - S_n = \)
(a) 0
(b) 1
(c) 3
(d) 2
Answer: (a) 0

Question. In an A.P of 99 terms, the sum of all the odd numbered terms is 2550. Then the sum of all 99 terms is
(a) 5039
(b) 5029
(c) 5019
(d) 5049
Answer: (d) 5049

Question. If the first, second and the last terms of an A.P. are \( a, b, c \) respectively, then the sum of the A.P. is
(a) \( \frac{(a+b)(a+c-2b)}{2(b-a)} \)
(b) \( \frac{(b+c)(a+b-2c)}{2(b-a)} \)
(c) \( \frac{(a+c)(b+c-2a)}{2(b-a)} \)
(d) \( \frac{(a+2c)(b+c+2c)}{2(b-a)} \)
Answer: (c) \( \frac{(a+c)(b+c-2a)}{2(b-a)} \)

Question. Four numbers are in arithmetic progression. The sum of first and last terms is 8 and the product of both middle terms is 15. The least number of the series is.
(a) 4
(b) 3
(c) 2
(d) 1
Answer: (d) 1

Question. If n arthmetic means are inserted between 2 and 38, then the sum of the resulting series is obtained as 200, then the value of n is
(a) 6
(b) 8
(c) 9
(d) 10
Answer: (b) 8

Question. If \( m > 1 \) and \( n \in N \) then
(a) \( \frac{1^m + 2^m + \ldots + n^m}{n} > \left( \frac{n+1}{2} \right)^m \)
(b) \( \frac{1^m + 2^m + \ldots + n^m}{n} < \left( \frac{n+1}{2} \right)^m \)
(c) \( \frac{1^m + 2^m + \ldots + n^m}{n} \ge 1 \)
(d) \( \frac{1^m + 2^m + \ldots + n^m}{n} \le 1 \)
Answer: (a) \( \frac{1^m + 2^m + \ldots + n^m}{n} > \left( \frac{n+1}{2} \right)^m \)

Question. Sum of the series \( S = 1 + \frac{1}{2}(1+2) + \frac{1}{3}(1+2+3) + \frac{1}{4}(1+2+3+4) + \ldots \) upto 20 terms is
(a) 110
(b) 111
(c) 115
(d) 116
Answer: (c) 115

Question. The first and second terms of a G.P are \( x^{-4} \) and \( x^n \) respectively. If \( x^{52} \) is the eighth term of the same progression, then n is equal to
(a) 13
(b) 4
(c) 5
(d) 3
Answer: (b) 4

Question. How many terms of the series 1+3+9+ ... sum to 364?
(a) 5
(b) 6
(c) 4
(d) 3
Answer: (b) 6

Question. If a, b and c are in G.P., then \( \frac{b-a}{b-c} + \frac{b+a}{b+c} = \)
(a) \( b^2 - c^2 \)
(b) ac
(c) ab
(d) 0
Answer: (d) 0

Question. If x, y, z are the three geometric means between 6, 54, then z =
(a) \( 9\sqrt{3} \)
(b) 18
(c) \( 18\sqrt{3} \)
(d) 27
Answer: (c) \( 18\sqrt{3} \)

Question. \( H_1, H_2 \) are 2 H.M.'s between a, b then \( \frac{H_1 + H_2}{H_1 . H_2} = \)
(a) \( \frac{ab}{a+b} \)
(b) \( \frac{a+b}{ab} \)
(c) \( \frac{a-b}{ab} \)
(d) \( \frac{ab}{a-b} \)
Answer: (b) \( \frac{a+b}{ab} \)

Question. If \( H_1, H_2, \ldots, H_n \) are n harmonic means between a and b (\( \neq a \)), then the value of \( \frac{H_1 + a}{H_1 - a} + \frac{H_n + b}{H_n - b} = \)
(a) n + 1
(b) n - 1
(c) 2n
(d) 2n + 3
Answer: (c) 2n

Question. If \( \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \ldots \infty = \frac{\pi^4}{90} \), then \( \frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \ldots \infty \) is equal to
(a) \( \frac{\pi^2}{36} \)
(b) \( \frac{\pi^4}{48} \)
(c) \( \frac{\pi^2}{72} \)
(d) \( \frac{\pi^4}{96} \)
Answer: (d) \( \frac{\pi^4}{96} \)

Question. The rational number which is equal to the number \( 2.\overline{357} \) with recurring decimal is
(a) \( \frac{2355}{1001} \)
(b) \( \frac{2370}{999} \)
(c) \( \frac{2355}{999} \)
(d) \( \frac{2359}{991} \)
Answer: (c) \( \frac{2355}{999} \)

In this section each question contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason).
Each of these questions has following four choices (1), (2), (3) and (4), only one of which is the correct answer.
1) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
2) Statement -1 is True, Statement-2 is True; Statement-2 is not a correct explanation for Statement-1.
3) Statement-1 is True, Statement-2 is false
4) Statement-1 is False, Statement-2 is True.

Question. Statement-1: If \( a_1, a_2, a_3, \dots, a_n, \dots \) is an A.P. such that \( a_1 + a_4 + a_7 + \dots + a_{16} = 147 \), then \( a_1 + a_6 + a_{11} + a_{16} = 98 \)
Statement-2: In an A.P., the sum of the terms equidistant from the beginning and the end is always same and is equal to the sum of first and last term.
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. Let \( a_1, a_2, a_3, \dots, a_{n-1}, a_n \) be an A.P.
Statement-1: \( a_1 + a_2 + a_3 + \dots + a_n = \frac{n}{2}(a_1 + a_n) \)
Statement-2: \( a_k + a_{n-k+1} = a_1 + a_n \) for \( k = 1, 2, 3, \dots, n \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. Statement-1: There exists no A.P. whose three terms are \( \sqrt{3}, \sqrt{5} \) and \( \sqrt{7} \).
Statement-2: If \( a_p, a_q \) and \( a_r \) are three distinct terms of an A.P., then \( \frac{a_p - a_q}{a_p - a_r} \) is a rational number.
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. Statement-1: If for any real \( x \), \( 2^{1+x} + 2^{1-x}, \lambda \) and \( 3^x + 3^{-x} \) are three equidistant terms of an A.P., then \( \lambda \geq 3 \)
Statement-2: A.M \( \geq \) G.M for
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. Statement-1: If \( x > 1 \), the sum to infinite series \( 1 + 3\left(1 - \frac{1}{x}\right) + 5\left(1 - \frac{1}{x}\right)^2 + 7\left(1 - \frac{1}{x}\right)^3 + \dots \) is \( 2x^2 - x \)
Statement-2: If \( 0 < y < 1 \), the sum of the series \( 1 + 3y + 5y^2 + 7y^3 + \dots \) is \( \frac{1+y}{(1-y)^2} \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. Let \( a, b, c \) be positive real numbers in H.P.
Statement-1: \( \frac{a+b}{2a-b} + \frac{c+b}{2c-b} \geq 4 \)
Statement-2: \( \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq 3 \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. Suppose four distinct positive numbers \( a_1, a_2, a_3, a_4 \) are in G.P. Let \( b_1 = a_1, b_2 = b_1 + a_2, b_3 = b_2 + a_3 \) and \( b_4 = b_3 + a_4 \).
Statement-1: The numbers \( b_1, b_2, b_3, b_4 \) are neither in A.P. nor in G.P. and
Statement-2: The numbers \( b_1, b_2, b_3, b_4 \) are in H.P.
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. Statement-1: If \( a, b, c \) are distinct real numbers in H.P, then \( a^n + c^n > 2b^n, \forall n \in N \)
Statement-2: A.M > G.M > H.M
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. Statement-1: \( \frac{1^2}{1.3} + \frac{2^2}{3.5} + \frac{3^2}{5.7} + \dots + \frac{n^2}{(2n-1)(2n+1)} = \frac{n(n+1)}{2(2n+1)} \)
Statement-2: \( \frac{1}{1.3} + \frac{1}{3.5} + \frac{1}{5.7} + \dots + \frac{1}{(2n-1)(2n+1)} = \frac{n}{2n+1} \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. Let \( n \in N \) and \( k \) be an integer \( \geq 0 \) such that \( S_k(n) = 1^k + 2^k + 3^k + \dots + n^k \)
Statement-I: \( S_4(n) = \frac{n}{30}(n+1)(2n+1)(3n^2+3n+1) \)
Statement-II: \( ^{k+1}C_1 S_k(n) + ^{k+1}C_2 S_{k-1}(n) + \dots + ^{k+1}C_k S_1(n) + ^{k+1}C_{k+1} S_0(n) = (n+1)^{k+1} - 1 \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (d) 4

Question. Let \( S_n \) denote the sum of \( n \) terms of the series
\( 1^2 + 3 \times 2^2 + 3^2 + 3 \times 4^2 + 5^2 + 3 \times 6^2 + 7^2 + \dots \)
Statement-1: If \( n \) is odd, then \( S_n = \frac{n(n+1)(4n-1)}{6} \)
Statement-2: If \( n \) is even, then \( S_n = \frac{n(n+1)(4n+5)}{6} \)
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. Statement 1: The sum of series
1+(1+2+4)+(4+6+9)+(9+12+16)+....+(361+380+400) is 8000.
Statement 2: \( \sum_{k=1}^n (k^3 - (k-1)^3) = n^3 \) for any natural number n. [AIEEE 2012]
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1